Even in framework days I nearly always found virtual directories caused more issues than they solved, so I'd never noticed this was even supported in v8 let alone that support had been dropped from v9. Though if I'm following it right, and from some playing about with a v13 site on my IIS server here, it isn't possible to get it working as a Virtual Directory, but Microsoft do say VDs aren't supported in .net Core and you should use Sub Apps by doing 'convert to application' on a physical subfolder instead. Doing it as a sub application instead, it does seem its possible to get the frontend of a simple v13 site working and routing pages correctly, as long as you have the culture on the root node setup to the first level folder url too. However back office doesn't work as it always looks to root of the domain for everything.
That said if the main issue preventing it working fully seems to be a lot of the Umbraco backoffice calls always look back to the root of the domain rather than the subfolder, causing subsequent problems with finding back office assets, then I'd have thought that same issue would exist even if you were reverse proxying it from a subfolder URL instead. You'd be able to reverse proxy frontend URLs from the folder fine and handle the generation of correct URLs yourself in the FE code. But you'd still need to have the back office side running on a real other domain so it could be found at a root of 'something'