I would guess that the pedagogical value of learning something like first order logic before undergrad algebra/topology/analysis is less about the specific content and more about getting used to and getting practice in formal expectations in math (specifically, advanced undergrad/early grad).

The expectation is that you need to take a semi-formal sentence formulated within a semi-formal language, set of axioms, and key theorems and generate a semi-formal proof about it. The motivation and context in which these sentences originally found value are missing, and the only mode of work that is available is this semi-formal setting. Therefore, you need to be ready to work without context and motivation while still making some kind of progress in understanding - which is what this semi-formal setting allows.

But group theory is unintuitive, and so people can have a hard time if they have to learn this formality AND group theory at the same time. On the other hand, logic (at this level) is pretty intuitive. We can reason about how things "should" be and so we're grounded in something familiar and building up the formal on top of it. And since you're already familiar with the structure of a theorem or how to use symbols to make a proof, you'll be able to focus on the group theory part of a group theory class.

So First Order Logic is a good way to train these skills. A different question would be whether or not we actually need Abstract Algebra to be done in this formalized setting? It's not like "real" math is just people solving dry textbook problems that have no meaning or motivation. A more pedagogically friendly context for Abstract Algebra could also fix the problem of people hitting a wall in the course, but it would take work from professors to try and think about different ways of teaching which is the last thing many of them want to do.