Roughly half of the error handling issues reported by PhpStorm have been fixed. The following categories of issues still need fixing:

1. Wrapper-level components implement the driver-level interfaces which are allowed to throw a driver-level exception but the consumers don't catch them because they are in fact impossible. This will be addressed in Do not implement driver-level interfaces by wrapper-level classes #4159.
2. The lazy platform detection via Connection::getDatabasePlatform() can fail at literally any point in time.
3. The "not supported" exceptions thrown in AbstractPlatform::get*DeclarationSQL() and similar methods should have a special type that could be declared as unchecked (e.g. LogicalException). Those are not meant to be handled at runtime (see the details).
4. Similarly to the issue addressed in Add TypeRegistry constructor #4145, there are issues with Type::getType() that technically may thrown an exception but practically doesn't and those cases are reported as not handling the exception (need a specialized exception-less API).
5. The Schema namespace depends on the Type one internally but doesn't handle its exceptions and therefore may throw a generic DBALException.
6. The wrapper-level components that wrap the driver ones should be more specific about the exception they throw and when possible declare it as the wrapper-level DriverException instead of the generic DBALException.