Here is one of numerous posts mirroring one of the module resolution issues that has plagued my project: https://stackoverflow.com/questions/39957100/visual-studio-2...
There are equivalent reports on GitHub and elsewhere on the net, as well as numerous related but separate issues.
IDEs from the same organization that sponsors TS don't even work reliably with it. Note one of the responses: "try VSCode." Indeed, VSCode works for me on the exact same code.
Currently I'm doing the same thing as you: running the TS output + the non-TS codebase through a giant blender of browserify + babel.
At one point, I would use --outfile, which makes TSC bundle all your files and dependencies. Neat, right? Nope - it gets the order of the dependencies wrong, which you find out later, at runtime. The maintainers on GitHub said "don't use outfile."
I totally understand your frustrations with VS, though most of the problems you mentioned seem to be related to the TS integration in VS, and not with TS directly. But issues like yours are the reason why I lost hope in VS. I try to use it as little as possible.
That doesn't mean some enterprising soul wasn't able to make it work, but it does mean it wasn't officially supported.