You get the Planck units by setting c, hbar, G, and Boltzmann's constant to 1. This is convenient for notational purposes but it has no inherent physical significance.

That isn't the only way to argue for a minimal significant distance. Arguably QM sets a strict ~60-70 digit precision limit on physical constants [1], beyond which you arguably can't differentiate between discrete and continuous theories, and so a minimum distance seems like a perfectly sensible way to frame it.

[1] https://physics.stackexchange.com/questions/16889/non-comput...