I have/have owned LF, FF, APS, 1", 2/3, and more. And I have used them all for macro at one time or another. Everything is a trade-off, but my preferred choice is the 1" sensor (16MP Nikon 1 V2) with a high quality macro lens adapted to it (Sigma 150/2.8).
With smaller pixels diffraction is more of an issue, but a crop sensor gives more DOF for a given composition/settings so you don't have to stop down so far.
Larger sensors do get more light, but with macro a big problem with light is the lens/camera/your shadow. With the crop sensor you don't have to get as close, which often lets more light reach the subject. It's also easier to use supplemental lighting when the lens is not in the way.
As above, you don't have to get as close with a crop sensor, which is a bit less prone to cause insects to flush/hide.
As above, you don't have to get as close, so you don't have to stop down as much for DOF.
Using a smaller sensor of the same resolution (different compositions) IS more resolution of the subject, which is the same end result as using a lens of greater magnification; assuming the lens can resolve more than the larger pixel sensor can. Even if the lens cannot resolve down to the level of the smaller pixels it does not necessarily mean less resolution. I.e. diffraction is a loss of recorded resolution, but that does not mean the result is less recorded resolution than a lower resolution sensor would achieve.
These are all single image captures taken with the Nikon 1 V2 + Sigma 150/2.8.
Orb Weaver Spider
Assassin Bug Nymph (?)
Juriniopsis (?, horsefly)
That is not to say I could not do better using FF, I probably could; but to do better using FF is very difficult... it would most likely require stacking a multiple of images, which introduces its' own challenges/issues.
I think the main difference is what he disclosed when he took the first picture with the APS camera... he used a cheaper/lower quality lens on the APS. Also note that I would call what he was doing "close-up" and not macro (≥ 1:1)