Is the truth table method for valuating 0th order sentences not a proof system in its own right?

This might sound a bit opinionated or a bit too pedantic, but in every book (that I’ve looked at) about propositional logic, usually this chain of events happens:

• Discuss the alphabet and grammar of 0th order logic
• Discuss how to evaluate a sentence’s truth value given an assignment (ex. True and True is True)
• Discuss the method of truth tables to show that two sentences are semantically equivalent – Here lies my problem, this method is not presented as a proof system in many texts I’ve seen.
• Discuss the notion of a proof system to help us deal with the intractability of truth tables

We usually say that prop. logic is sound and complete but hear me out: These properties depend on two specific "methods" (for a lack of better word) – The first is the truth table method, and the other is the proof system we choose. Perhaps we choose truth tables because they’re clean and easy to understand? Truth tables are nice, but don’t extrapolate well for more complicated logics.

It turns out there’s a lot of equivalent proof systems for prop. logic. So let’s suppose we pick one and put it on the "semantic" pedestal. Call it X. Now we pick another one, Y, and call it our "proof system". Now we can talk about whether these two are sound and complete again, because even though they’re both proof systems, we chose one to be "what we mean" by a sentence.

So hence the question: Why do we not introduce semantic evaluation of statements as a proof system itself? Is it because tables are too cumbersome in general to describe in terms of inference rules?