If you use handle_path to strip the path prefix, then your upstream app needs to be aware of that path prefix to properly build paths. If the app doesn’t have somekind of “base path” config, then you shouldn’t use subpaths like this, and using subdomains for each part is a better option.
The nginx config is only proxying to the one port, shouldn’t you just do that as well? Just have one reverse_proxy with no matcher.