Do C++ modules solve this problem?

Kind of, the current problem is they aren't fully mature, you great a good experience with VC++ and clang (vLatest), alongside MSBuild/CMake/ninja.

VS IDE tooling is stil kind of broken because they rely on EDG instead of VC++, and fixing module intelisense has been low priority for EDG, and MS as well as I would assume a $4 trillion valued company would care about their suppliers.

Clion has better support on that regard.

GCC is still not fully there, and Apple clang, well Apple is happy with module header maps for their Swift/Objective-C integration, so lets see.

C++ modules solve exactly one of the problems - the "one pch limit" one - at the cost of introducing several more. Certainly they are not more compiler-independent!