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Interest in alternative systems never dies. Although ZF-style set theory (or more precisely in my opinion, "cumulative-hierarchy-style set theory") is by far dominant, there's no inherent reason for that to remain the case forever, and there's certainly no reason to abandon the study of alternative set theories in general. Limitation of size does play an important role in such theories, so I'd say that the answer to your question is a weak "no."

However, I think this also misses the broader question: why did limitation of size (at least as such) fade away in the first place? We have to understand that before we decide what role limitation of size should play in the next set theory we cook up.

First, we remember that there are really two pieces to limitation of size. The first is that any class which surjects onto the universe of sets is a proper class. You mention that limitation of size is arguably too strong; well, this half of limitation of size is too weak to be useful on its own (although it's an important motivating force - e.g. behind replacement). The worryingly strong direction is the converse, which says that any class which does not so surject is a set.

The intuitive point now is that essentially as long as we have regularity and enough replacement - and I'll call this "cumulative-hierarchy-style" class or set theory - we can show that any proper class surjects onto the ordinals. Namely, sending $x$ to the rank of $x$ gives a surjection onto a cofinal class of ordinals, and the Mostowski collapse turns this into a surjection onto the whole class of ordinals. So by composing surjections, limitation of size holds iff there is a surjection from the ordinals to all sets. This in turn is equivalent to the existence of a well-ordering of the universe of all sets, aka global choice.

Now the key point is the above makes sense in mere set theory (in particular, ZF). Of course, on the face of it that's nonsense since we talked explicitly about classes, which we can't do in ZF. Instead, in ZF everything is about (parameter-)definable classes. But the above argument still essentially goes through, and we can prove that given a model $M$ of ZF (or indeed much less), if every parameter-definable class in $M$ is either a set in $M$ or definably surjects onto $M$, then there is a parameter-definable surjection of the $M$-ordinals onto $M$, and the converse holds as well.

It turns out that this can be collapsed into a single first-order(!) statement: namely, that there is some set $A$ such that every set is definable from $A$ together with an ordinal. (This isn't hard to see - we just say "$x$ is the $\alpha$th element of the well-ordering of $V$ induced by $s$," where $s$ is our $\{A\}$-definable surjection from the ordinals to $V$.) This can be written as "$V=$ HOD[A] for some set $A$." In case we have a parameter-freely definable surjection from the ordinals to $V$, we get $V=$ HOD. Just like in the case of the axiom of constructibility it's not immediately clear that this is actually first-order expressible, but a neat trick with the reflection principle shows that it is. So limitation of size for definable classes is a first-order principle even in mere set theory (or at least, in ZF). Now HOD and its variants are extremely important concepts in modern set theory even ignoring foundational considerations, so the "HOD-language" tends to win out (and certainly wins out when looking at ZF or its extensions).

The final piece of this picture is the shift in interpretation. Initially we may have thought of limitation of size as a maximizing principle (anything that could be a set, is), but in light of its equivalence with $V=$ HOD (ignoring parameters for now for simplicity) it threatens to take on the opposite character in cumulative-hierarchy-style set theory: which is more restrictive, that every set have some ordinal defining it or that there be no ordinal which lets us define some fixed set? The cumulative hierarchy idea pushes against the "obviously maximizing" nature of limitation of size. So it's hard to justify using limitation of size as a centerpiece of a set theory if we're committed to the centrality of the cumulative hierarchy idea and to the value of maximization of mathematical concepts, and these seem more deeply entrenched.