TypeScript: Update tooling or remove

[pago]

Explain the problem

Our current TypeScript integration is rather limited and dated. We should decide whether we want to keep using it (in that case: Switch to Babel TypeScript integration + ts-check) or if we consider it to be a failed attempt.

My personal preference would be to double down on TypeScript and to start converting our code base to fully utilize it. Especially a complex project like Melody which involves a parser, compiler and various transforms could benefit from type information and might be more approachable if we have that available.