Your suggestion would purposely share the same exact private torrent infohash into public DHT/PEX,
Yes, but not necessarily. It is trivial to recompute the infohash with the private bit disabled. This would split the network, but that is probably a good thing to preserve the anonymity of the private tracker users, as pointed out by another commenter.
Yes, with this solution, internally it could be seen a two separate torrents, but if it is an option easily accessible in the client settings, and it is handled transparently as a single torrent, much more people will do it, and the scene as whole would gain with the network effect.