This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little:
Is it not more natural consider NBG set theory as the foundation for modern algebra as opposed to traditional ZFC? To me, ZF has always seemed sort of hacky, for lack of a better word, as if it has been patched and patched over the years; kind of how windows vista would look today if it were still in use. It is no doubt an extremely powerful theory, but the point is that in modern application ZF tends to be somewhat inadequate, seemingly always requiring a work around; thus, hacky. On the other hand, NBG deals with classes directly, and is just for all intents and purposes more accessible from the algebraic viewpoint, especially from the point of view of lattice and order theory, all the way to class field theory. NBG is just better equipped for the job.
I guess an easier way to say all of this is that while ZF is more concerned with objects, NBG is designed to exploit the relationships between objects, which, in my opinion is more fundamental to not only mathematics, but to logic itself. NBG is implemented naturally to exhibit the abilities of comparison and deduction, which can be argued to form the basis for the concept of logic, in and of itself.
Am I crazy, or has anyone else ever felt this way?
Many people have felt like that.
This is why you often hear people in algebra moaning about set theory, or ignoring set theoretic issues (they usually can point out where these issues arise, and that someone out there knows how to solve them). This is also why there are people who are very enthusiastic about algebraic set theories like $\sf ETCS$, or type theories like $\sf SEAR$ and $\sf HTT$, which may or may not prove to be a better foundation for algebra.
But switching from $\sf ZFC$ to $\sf NBG$ only pushes the problem "one step further". Sure, now you have the class of all groups as an actual object. But what about the category of all small categories? That's not a class anymore, since only sets are allowed to be elements of other classes, and small categories are not necessarily classes.
This is why working with universes is easier here. They allow you to jump "one level up" without any consequences. Each time you just extend the definition of what it means to be a set, and include more things as sets.