1. It seems that they write a ".d.ts" file manually in addition to using TypeScript for the code. This is dumb, since TypeScript generates the declarations automatically. However, for them "it was too much overhead and complexity when we attempted it before", which caused them to give up, a very dubious course of action.
2. They claim that changes take minutes to recompile, but TypeScript can compile incrementally, so this shouldn't happen assuming they are organizing their code properly. Also, you can just translate without type checking, which is no worse than using JavaScript instead.
3. They claim that "having two Body classes is obviously wrong" (?!). Of course having two classes with the same name in different namespaces/packages is perfectly fine in properly designed languages (including both JavaScript and TypeScript). Their "Header" shadowing problem also might be due to a lack of understanding of namespaces.
4. They seem to conflate JavaScript vs TypeScript with single file vs multiple files. Of course you can have a single TypeScript file or you can have multiple JavaScript files and bundle them with any JS bundler or just concatenate them.
Given that their codebase seems to be complex according to their compile time claims and that they don't seem particularly skilled given their claims, using a type-deficient language will most likely result in software full of bugs.