Recall Muller's criteria in Sets, Classes and Categories: page 14 for a theory that founds Mathematics.
Arn't those captured simply by $$\sf Zermelo + \text {worldly cardinals exist} $$
define sets as classes with ranks lower than the first worldly cardinal and everything would go through!
A worldly cardinal is some cardinal $\kappa$ such that $V_\kappa\models \sf ZFC$