The reported crash cannot be reproduced in ArcGIS Pro 3.3, 3.4, or 3.5.
The described behavior is expected and as-designed. Publisher and administrator connections are no longer allowed to an ArcGIS Server stand-alone deployment configured to use HTTP-Only communication. When HTTP-Only communication is used, usernames and passwords associated with built-in authentication accounts are not encrypted when requests are sent across the network. This change was made to avoid ArcGIS Pro from being a security risk by exposing credentials with elevated permissions on the server to possible attackers.
BUG-000163598 is an issue that seeks to update the ArcGIS Pro help system with information regarding the expected behavior when accessing an ArcGIS Server deployment configured for HTTP-Only communication.