It doesn't matter what they believed. Philosophy never involves rigorous proofs. By adding them you would just end up doing math. A partially finished proof is still math and not equivalent to 20 pages of worthless babble about human understanding. Throwing darts at the page and putting equations where they land will not change that. Philosophers constantly pull from the same small bag of tricks - inserting "science" or "philosophy of <science>" or "meta<science>" into their titles, Sokaling in random disconnected bits of scientific terminology to sound more credible, and trying to claim criticizing philosophy is philosophy to avoid criticism. It's unconvincing and embarrassing to hear from the self declared intellectuals responsible for some of the biggest false beliefs about science in history.
If by "philosophy" you mean work that not only lacks a rigorous proof, but isn't even a step in the direction of a rigorous proof, you'll be happy to hear that many philosophers - sorry, mathematicians who mistakenly consider themselves philosophers - share your opinion of it. When I said "philosophy" I was referring to the academic field, which includes a lot of work that you consider math. While I think complete non-mathematician philosophers like Deleuze have value in their own way, I certainly wouldn't call them rigorous or useful to modern science.
I'm not clear on whether you think The Concept of Truth in Formalized Languages falls into the "actually just mathematics" category or the "making up random equations" category. If the latter, I assure you that Tarski's proofs are sound. Here's a simple explanation of the most famous result from the paper in case you found the original proof inaccessible: https://qubd.github.io/files/TarskiUndefinability.pdf. A more general discussion of Tarski's work and other axiomatic theories of truth can be found at the Stanford Encyclopedia of Mathematics: https://plato.stanford.edu/entries/truth-axiomatic/
It doesn't matter what you consider yourself. Or who is doing to considering, despite your efforts to emphasize that.
The proofs are math. We've already established that math is sound. This discussion is not about the merits of math, we're talking about philosophy. Things like "The transfer of understanding from one person to another is not automatic. It is hard and tricky. Therefore, to analyze human understanding of mathematics, it is important to consider who understands what, and when." are philosophy. It's not difficult to separate, you're just trying to make it seem like it is to blur the lines between a pseudoscience and actual science. Again, disguising worthless philosophical ramblings with mathematics doesn't make your philosophy any more useful.
I am focusing on mathematics because I am more familiar with mathematics than philosophy and dislike seeing it misrepresented, particularly to use as a cudgel against a related field that I respect.
The passages that you describe as "worthless philosophical ramblings" are part of Tarski's results. He could not have left them out without obscuring the meaning of his proofs. Possibly model theory would not exist today had he done so. It would certainly have taken longer to develop.
Another instructive example is Per Martin-Lof's lectures On the Meanings of the Logical Constants and the Justifications of the Logical Laws: https://www.ae-info.org/attach/User/Martin-L%c3%b6f_Per/Othe.... Unlike Tarski's paper, this contains no formal proofs whatsoever; if the sentence you quote is worthless, then I imagine "There is no evidence outside our actual or possible experience of it. The notion of evidence is by its very nature subject related, relative to the knowing subject, that is, in Kantian terminology." is worse than worthless. Nevertheless these lectures have been of great importance in logic and computer science. You can see some of their impact in the citations here: https://scholar.google.com/scholar?cites=2483744927635326348
You may be unable to find any useful meaning in this kind of writing, but most mathematicians do not share your difficulty. This is fortunate, since the field would be greatly impoverished if it purged itself of all philosophy and philosophy-adjacent work. I would normally encourage you to read https://terrytao.wordpress.com/career-advice/theres-more-to-... on the role of non-rigorous big picture thinking in mathematics, but it deals entirely with human understanding of mathematics and is therefore only of interest to Terence Tao and other such pseudoscientists.
