Solving Ax=b is like week 2 of undergraduate linear algebra. Solving Ax+e=b, in a lattice setting, is Hard in the same sense factoring is: as you run the steps of elimination to attempt to solve it, the errors compound catastrophically.
What you described would be closer to learning with errors (https://en.m.wikipedia.org/wiki/Learning_with_errors) combined with SIS/SVP. Learning with errors is based on the parity learning problem (https://en.m.wikipedia.org/wiki/Parity_learning) in machine learning, which I take as a positive sign of security.
Interestingly you can get information theoretically secure constructions using lattices by tuning the parameters. For example, if you make the `A` matrix large enough it becomes statistically unlikely that more than 1 solution exists (e.g. generate a random binary vector `x` and it's unlikely that an `x'` exists that solves for `b`).
I found https://er4hn.info/blog/2023.12.16-sphincs_plus-step-by-step... to be a nice introduction.
I found AI (combo Grok & ChatPPT 4o) to be the best resource for this. It was able to break it down to digestible chunks, then pull it together that made sense. It even made suggestions what math areas I need to brush up on.