A possible answer to the last part of the question: the article Six Easy Roads to the Planck Scale, Adler, Am. J. Phys., 78, 925 (2010) contains multiple "derivations" that you might (or might not) find more satisfactory than the one you mention.
As far as the rest of the question is concerned, others have made the most relevant points. I think a fair summary of what Magueijo is getting at is something like the following:
One frequently hears that "interesting new physics" happens when some length $l$ is less than the Planck length. The Planck length is manifestly Lorentz invariant. The other length $l$, if it is the physical length of some object, is manifestly not Lorentz invariant. What meaning, then, can one assign to such statements?
It seems to me that reasonable people can differ over whether this is an interesting question. I don't find it manifestly insane, myself.
There are many, many instruments which are calibrated using the old definition of the kilogramme - the International Prototype Kilogramme (IPK) made of a platinum-iridium alloy.
So one kilogramme measured using the new definition had to be as close as possible to one kilogramme using old definition so as not to have to recalibrate all instruments which relied on the old definition of the kilogramme.
Using the old definition of the kilogramme (IPK) the numerical value of Planck’s constant was measured as accurately as possible using the Kibble (watt) balance and the X-ray crystal density method.
The two values that you quoted $6.62606957 \times 10^{−34}\, \rm kg\,m^2s^{-1}$ and later $ 6.62607015 \times 10^{−34}\, \rm kg\,m^2s^{-1}$ were the results of such measurements of Planck's constant.
As of 20 May 2019 the determination/definition was turned on its head with the value of Planck’s constant defined as $ 6.62607015 \times 10^{−34}\, \rm kg\,m^2s^{-1}$ and the IPK (made of a platinum-iridium alloy) having a measured value of one kilogramme to within one part in $10$ billion.
On page 131 of the BIPM brochure on the SI system of units it states:
The number chosen for the numerical value of the Planck constant in this definition is such that at the time of its adoption, the kilogram was equal to the mass of the international
prototype, m(K) = 1 kg, with a relative standard uncertainty of $1 \times 10^{-8}$, which was the standard uncertainty of the combined best estimates of the value of the Planck constant at that time.
In future the new definition of one kilogramme via the defined exact value of Planck's constant will enable measurements to be made to see by how much the masses of the IPK and its daughters change with time.
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why did they choose this arbitrary value of 6.62607015×10−34.
This value was chosen to make one kilogramme using the old definition (IPK) as close as possible to one kilogramme using the new definition (via Planck's constant).
instead of something more exact like 6.62607x10-34.
This would have required the recalibration of many, many (accurate) instruments.
Best Answer
The Planck force can be interpreted with general relativity only, without need for quantum mechanics.
It is roughly the force between two black holes of mass $M$, located at each other's event horizon, i.e. at a distance given by the Schwarzschild radius $R=\frac{2GM}{c^2}$. Of course Newtonian mechanics is not applicable here anymore. But we can still use it to get the order of magnitude for the gravitational force between the two black holes: $$F=\frac{GM^2}{R^2}=\frac{c^4}{4G}$$