I can definitely buy a lack of understanding on my part - my understanding of all this is quite fuzzy. But I'm not sure string theories reproduce existing theories so much a say this is compatible with the other stuff in a handwavy way? I don't think there's any "we assume strings and thus can deduce quantum mechanics and general relativity" stuff? I don't think anyone can deduce quantum mechanics and Einstein's deductions with relativity are based on simple observations like the speed of light being observed to be constant and acceleration seeming similar to gravity?

It's not "we assume strings and deduce QM" it's "strings are a model which is defined so as to produce QM and also...". It is, by design, a theory of QM and GR. That's the point. It's a claim about what lies "below" the level of the individual fields in QFT which gives rise to them (and also gravity).