Hello st_123,
According to the ArcGIS (Desktop, Engine, Server) Support for Oracle 19c Patch website, the only requirement for the patch is that the ArcGIS product be installed on your system before you can install the patch. I wasn't able to find any information on the patch causing ArcMap to crash.
Since database connections work correctly in ArcCatalog, that leads me to believe the Oracle 19c installation/setup is successful.
If it were me, I think my next step would be to uninstall/re-install (or repair) ArcMap 10.7.1 to see if that solves the crashing issue. Just be sure to backup any map documents and data before doing so.
If that doesn't work, you could try your idea: "Or should I uninstall the 12c first and then install 19c oracle, then to installed the 19c patch for ArcMap 10.7.1.". Before doing so, I'd make sure to do research on the potential impacts of uninstalling Oracle 12c and make your decision from that.
I don't believe there is a problem with the sequence here, but you never know.
Cheers,
Mike