There's a more basic problem with string theory, which is that it's not a theory. It's a mathematical framework which is compatible with a very wide range of specific physical theories.
About tests of quantum gravity, there have been proposals for feasible tests using gravitationally-induced entanglement protocols:
I don't think that's quite the problem. In mathematics, the word "theory" is often used when referring to particular mathematical frameworks (e.g. Group Theory, Graph Theory, Morse Theory). In that sense I think String Theory is very much a theory. As you imply, in physics, the word "theory" is typically used in a different sense. I'm not a physicist but I presume a physical theory has to be verifiable, consistent with observations, able to predict the behavior of unexplained phenomena. If I understand correctly, the basic problem is that in some quarters string theory is being passed off as a physical theory. I know of pure mathematicians who are interested in string theory and who couldn't care less whether its a physical theory.