It's my understanding that the invention of the metric system during the turbulence following the French Revolution also included a switch to decimal time, with ten hours per day, etc., but that it didn't take. There's a certain amount of cultural inertia that has to be overcome; as you're probably aware, those of us in the United States still have many miles to go before we can fully adopt the metric system.
As you say, you have to give anonymous inventors of the 24-hour day credit: while the metric approach of powers-of-ten relationships between units is dreadfully easy to handle when you're using base 10 arithmetic, it's quite difficult to divide ten things into three equal-size sets. Remember that base ten is essentially an arbitrary choice made because most people have ten fingers and spend their childhood grouping things into fives and tens to count them.
Twenty-four has boatloads of divisors: you can separate into a dozen pairs, three groups of eight, or six quartets. Sixty would make a pretty nice base, since it's the first number divisible by two, three, four, and five; but sixty is too many things for most people to count in their heads.
The second is actually historically based not on the length of a year, not of a day:
until the adoption of the cesium clock standard in 1960, the definition of the second was actually the appropriate fraction "of the tropical year 1900." It took roughly half a century for the standards committee to realize that we can't go back and re-run the year 1900 to see whether we're still producing correct seconds.
There are several things that the SI system does that don't quite make as much sense as you might like. Why on earth does the base unit for mass, the kilogram, have a prefix? Why is the base unit for electricity the ampere, when we've known for a century that charge occurs naturally in standard-sized lumps? I put the SI endorsement of the historical relationship between the second, the minute, the hour, the day, and the year in the same category. It's a convenient unit with strong historical and popular support. I don't see a need to decimalize the day.
Emilio Pisanty asks for references.
The Time Service Deptartment at the U.S. Naval Observatory, which is responsible for inserting leap seconds every ~500 days to keep atomic time (as defined) from slipping relative to ephemeris time, seems to be the source for the Wikipedia account of the history of the second, but does not cite additional sources.
The Bureau International des Poids et Mesures, which is the orgainization responsible for defining and revising the international system of (SI) units, does not discuss the history of the second in its brief history of SI, and does not seem to have a page describing the history of the second in the same detail as the history of the meter.
The NIST/CODATA reference website contains historical background statements about the different fundamental units; the page for the second describes the shift from ephemeris time to the cesium standard as above.
An abandoned-looking, authorless site about units repeats the same story, but includes a handful of technical and non-technical references, including a 1958 article (Markowitz et al.) entitled "Frequency of Cesium in Terms of Ephemeris Time." This reference discusses plans at that time to move to the atomic standard.
The articles citing Markowitz et al. include a 2005 review in Metrologia entitled Atomic time-keeping from 1955 to the present, a much more detailed discussion with about four dozen techical references. A more recent review, Evolution of timescales from astronomy to physical metrology, seems from its abstract to offer a broader historical perspective.
For historical timekeeping systems and the decimal time adventure of the French Revolution
I happened across Carrigan, "Decimal Time", 1978, which cites
For the division of the day in 24 hours by Egyptians, and the 60x60 subdivisions of the hour by Babylonians: O. Neugebauer, The Exact Sciences In Antiquity, Brown University Press, 1957.
For a catalog by Hipparchus (ca 140 BC) of stars whose rising is separated by one-hour intervals, accurate to about one minute: the "time" article in the 11th edition of Encyclopedia Brittanica. The corresponding article in Brittanica online is quite lengthy, but hidden behind a paywall for me.
For a medieval division of time into lit and dark "tides" (in English, "noontide" and "eventide"), each with twelve "hours" but only having equal length near the equinox: K. Welch, The History of Clocks and Watches, 1972.
For a similar Oriental system not supplanted until Western commerce became important in the 1800s: J. Arthur, Time and its measurement, 1909.
Old papers have old references! Carrigan observes that while weights and measures are important enough for commerce that many local standards arose more or less at once, early precise timekeeping would be complicated by the vagaries of travel by ship or by land. The engineering skill to build a clock with a useful second hand "preceded to some extent the need for standards of communication at small time intervals[, which] may have led to the universality of the present time system."
Disclaimer: due to limited connectivity from where I'm now, I can give only a short answer and I'm not able to access useful references.
With the new SI the distinction between base and derived quantities (and units) will lose a lot of its foundational value, and will be kept mostly for historical continuity, and thus, as far as I know, there is no plan to change the set of base units (which would have to be anyway 7, to avoid changing the relationships between the physical quantities) or to get rid of them.
For what concerns the realization (not implementation) of the ampere with single electron transistors (SET), notice that at present the level of accuracy of such realizations is much worse than that achievable through the other path (Josephson effect plus quantum Hall effect), and it's insufficient for primary metrology. In fact, at present, the so called quantum metrological triangle has not yet been closed with the highest level of accuracy.
Best Answer
As is the case elsewhere in metrology, the answer is tied up in history, measurability, practicality, repeatability, past misconceptions, and consistency (despite those past misconceptions).
The history of atomic mass and the mole (the two are quite interconnected) goes back to the early 19th century to John Dalton, the father of atomic theory[1]. The unified atomic mass unit is named after him. Scientists of that era were just learning about elements; the periodic table was 60 years in Dalton's future. Dalton initially proposed using hydrogen as the basis. Issues of measurability and repeatability quickly cropped up. So did mistakes. Dalton, for example, thought water was HO rather than H2O[2].
These issues resulted in chemists switching to to an oxygen-based standard based on the oxygen found on Earth. (That elements can come in multiple isotopes was not known at this time.) Physicists' investigations at the atomic level caused them to develop their own standard in the 20th century, based on 16O rather than the natural mix of 16O, 17O, and 18O (atomic masses: 15.994915, 16.999131, and 17.999161, respectively, with a nominal mix of 379.9 ppm for 17O, 2005.20 ppm for 18O, and the remainder 16O) used by chemists.
The natural mix of the various isotopes of oxygen is not constant. It varies with time, place, and climate. Improved measurements and more widespread usage made repeatability become a significant issue by the middle of 20th century. The primary cause is natural variations in the two most common isotopes of oxygen, 16O (the dominant isotope) and 18O (about 2000 parts per million, on average). The IUPAC Technical Report[4] on atomic weights of the elements lists the atomic weight of naturally occurring oxygen as varying from 15.99903 to 15.99977.
The primary cause of these natural variations is the preferential evaporation and precipitation of water molecules based on various isotopes of oxygen. Water based on 16O evaporates more slightly readily than does water based on 18O, making tropical oceans a bit concentrated in 18O compared to average. On the flip side, water based on 18O precipitates slightly more readily than does water based on 16O. This makes precipitation in the tropics have slightly higher 18O concentrations compared to nominal, and it makes precipitation in high latitudes have slightly lower 18O concentrations compared to nominal.
Physicists had a solution: Switch to their isotopically pure 16O standard. This would have represented an unacceptably large change (275 ppm[3]) in chemistry's oxygen-based standard. It would have required textbooks, reference books, and perhaps most importantly, the recipes used at refineries and other chemical factories to have been rewritten. The commercial costs would have been immense. It's important to keep in kind that metrology exists first and foremost to support commerce. Chemists therefore balked at that suggestion made by physicists.
The carbon-based standard represented a nice compromise. By chance, defining the atomic mass as 1/16th of the mass of a mole of oxygen comprising a natural mix of 16O, 17O, and 18O is very close to a standard defining the atomic mass as 1/12 the mass of a mole of 12C [3]. This represented a 42 ppm change from the chemists' natural oxygen standard as compared to the 275 ppm change that would have resulted from changing to 1/16 of the mass of a mole of 16O [3]. This new standard was based on a pure isotope, thereby keeping physicists happy, and it represented an acceptably small departure from the past, thereby keeping chemists and commerce happy.
References:
Britannica.com on John-Dalton/Atomic-theory entry
I'm leary of referencing wikipedia. Britannica is still fair game for basic facts.
Class 11: How Atoms Combine
Dalton's mistake on assuming water was diatomic is widely reported. This is one of many sites that make this claim on Dalton's mistake.
Holden, Norman E. "Atomic weights and the international committee–a historical review." Chemistry International 26.1 (2004): 4-7.
I found this after the fact, after Emilio Pisanty asked me to find some references. This says everything I wrote, only better, in more detail, and with lots of references.
Meija, Juris, et al. "Atomic weights of the elements 2013 (IUPAC Technical Report)." Pure and Applied Chemistry 88.3 (2016): 265-291.
See table 1, and also figure 6.