The video shows someone doing large numbers of digits. This is a base-like calculation.
Computers (ie people who can do rapid calculations), can do this with some ease, but it takes practice. I can do several places of criss-cross calculations in various bases, like 120, and 10, but the mind does not hold the digits easily.
I'm pretty sure that some of the super feats done by the Trachenberg system gives the same sorts of results. In any case, the person might have a greater difficulty with a different base, because, like the chinese stone-cage, the process uses features of base 10.
My grandfather could add money columns (£,s,d) as fast as one can run a finger down the column. I suppose, it's not so much the mathematics, or the form of the number, but the kid in the video can just hold hundreds of digits and do lots of simple arithmetic slides.
On Bases
The only 'real' numbers are the digits. Anything else is represented as a path of remainders. So for example, '7' is a real thing, but '73' is a pathway to a number. The same numbers in dozenal are '7' and '61'.
Something like '7!' is a different path to a number. One can for example, learn 7! in ten or dozenal or 120, to get eg 5040, 2E00, and 4200. But these are not 'conversions' or even calculations. They're the outcome of rote-learnt tables. Other numbers do not lend themselves to such fast conversion. A similar-sized 5120, i do not know its dozenal form, but its twelfty form is 4280. I suppose you can write it as 5040 + 80, and this allows one to directly write 2E68 for the dozenal.
In all of the examples above, the numbers represent a series of remainders, but i did not go through the remainders to find the number. I saw 5040 as 7!, and wrote 7! in the various bases.
Some numbers i know only in base 10, some in base 12, some in base 120. The order of a gosset's E8, is 3.43.24.00.00. If you know further that this is 1.72*10!, you could work it out in dozenal, by noting that 10! = 1270000, and that 127 is dec 175, and this divided by 9 gives 19 4/9. From this one gets 1754 00000. The decimal, you could get by multiplying 6912 * 1008 * 100, which i think is 696729600.
Many years ago, i did a project to find the index of primes 2-19 for the primes: this is for a primitive root g, then eg $g^i=2 \pmod p$. This of course, suited the factor-model, because you would see something like, eg 507, and you say, $3.13^2$, and the like. I found it hard going when $p$ got to 56.00 (6720). I could not look at a number like say 3135 and say immediately, that it is 3*5*11*19. (this one i could, but there are others i couldn't that had fairly small divisors.) The factor method is not really something one does.
Some features are learnt through the fingers. A typist might tell that a word is typed wrong because the feeling from the keys is that. I know more that eg 696729600 feels right, rather than looking at the digits. Likewise, i can type in the various short-chords of the polygons, (the chord which makes the third side of a triangle of two edges), usually without thought, eg 1.801937736 or 1.93185165259 for {7} and {12}. I have typed these in quite often. The heptagon was done on a ten-digit calculator, the dodecagon is older, was done on a 12-digit calculator.
This is more like a comment but it's too long, so I'm putting it as an answer instead, please accept my apology.
In base $10$, the "symbol" $78152_{10}$ represents the number
$78152_{10}=7\cdot 10^4 + 8\cdot 10^3 + 1\cdot 10^2 + 5\cdot 10^1 + 2\cdot 10^0$.
In base $n$, the "symbol" $78152_n$ represents the number
$78152_n=7\cdot n^4 + 8\cdot n^3 + 1\cdot n^2 + 5\cdot n^1 + 2\cdot n^0$.
You can see that we prefer to call a base by the number that is raised to the power of its position from the last digit. That is why we call it the base.
Best Answer
The idea of a "base", and even the idea implicit in it of a consistent positional numbering system, is a relatively modern one. For that matter, even "base 10", in the sense of decimal numerals — the Hindu–Arabic numeral system we all use today — is relatively modern; witness the fact that much of Europe didn't begin to use it until well into the second millenium. The idea that we can count in any base is even newer.
Your argument might make sense if humanity started with the idea in mind of using a base-$b$ representation for some $b$, and then looked to their fingers to decide what the base $b$ should be. This of course is not what happened, nor even is it imaginable in any culture. The concrete precedes the abstract.
Instead, what we see historically is counting on one's fingers, and thus counting by fives or by tens, not counting in base 10. If you're counting on two hands, when you reach $10$ you literally run out of fingers to count, and that is where you have to leave a mental (or physical) note to yourself that you're done with one round of counting, and begin counting anew. (Similarly $5$, if using one hand.)
We can see traces of this "counting by $5$" or "counting by $10$" in early systems like Roman numerals: note that $8$ is represented as "VIII", denoting one count of five (done with one hand's worth of counting), and starting again, reaching up to $3$ in the process. Similarly, the representations "XX" and "XXX" show that they were being thought of as "two tens" and "three tens", rather than as "in base $10$, three in the tens place and zero in the units place" — the idea of base $10$ is not actually present. Thus "XXXVIII" literally denotes the process of counting: "three tens (three double-hands), a five (one hand) and three fingers". (There are even traces of counting by $20$s; consider the English "score" and French number names).
It was only after centuries of already counting by $10$s, conducting transactions with $10$-based words for numbers and so on, that the base-$10$ representation arose, as a representation system for the same numbers that everyone was already accustomed to thinking of in tens, and (very slowly) spread across the world.