It's the worst! I'm still testing my tool and I have to delete, republish, re-add the new service to a map EVERY time I make an adjustment. It seems pretty straightforward to me - If I can publish something (tool, map, layer, service, whatever) I wan't to be able to update it without deleting it.
I apologies for any convenience. At this moment, you need to
This functionality will be available in Pro 2.6, or the release right after that. Stay tuned.
The steps you listed are exactly what Lauren said she's doing.
And we have been 'staying tuned' - for over 2 years!
It is baffling as to why this core functionality wasn't included in Pro 1.0.
https://community.esri.com/thread/195618-possible-to-edit-gp-service-after-publishing
Past comments about when this bug will be fixed suggest that "or the release right after that" might be a long ways off.