Certainly, there are times where that's not reasonable. Maybe the core codebase is on a proprietary technology. Maybe it's built around an EoL framework. Maybe it's built on on a FOTM language that you can't hire for at all anymore.
In those cases, which are more rare than developers like to admit, I think a piecemeal migration makes sense. In my experience, it's better to do it by altering the consumers. Whether it's a frontend that can point to another endpoint, or an API gateway that can switch out the service it's pointed to, or making a shim layer (proxy), that can serve that purpose.
Having a service that is both the proxy and the new application is a poor separation of concerns, very difficult to reason about, and makes it too easy to intermingle the logic between the old and the new. In my experience.