what actually happens in the scenario of flying an approach in heavy rain?
Yer get wet.
My Aussie ATPL notes are about 10,000km away.
Practically wise, just keep on profile with the instruments and hope like heck you can see the PAPIs when visual.
To be honest, it's the touchdown and braking that's more of a hassle WRT heavy rain.
However,
Airbus has something to say about it.