https://github.com/bitcoin/bitcoin/pull/17509 introduces a Load PSBT menu option. It would be even easier if a user can double-click on a PSBT file, open it from an email, etc. For this we would need to register a file handler.
This can be done by defining a QAbstractFileEngineHandler for .psbt files, which should then call the same code as "File" -> "Load PSBT".
One potential downside of a file handler, is that it encourages users to double-click on potentially untrusted files, which might be nasty executables.