At that same company, we switched to Jira from 6 or 7 different ticketing systems (a mix of in-house and third party systems split up by department). It was a herculean effort to transition, but the result was much better than any of the individual ticketing systems with the huge benefit of just opening a ticket anywhere and moving it to where it needs to go while preserving the history.
Now the cons I've seen are; I have no idea how much it costs and it seems to require a small to large team to manage and deploy it for maximum usefulness. For ad hoc deploys I've heard/seen performance issues if the server isn't beefy or configured correctly and the UI can be inscrutable if users are just dropped into the default. That's about (the best case scenario of) what I expect for enterprise software, though.
I have used GitHub Enterprise and haven't used BitBucket. I might not be using it correctly, but I don't see what's so great about GitHub Enterprise (especially for the cost). We still have to use a separate ticketing system for non-project related things (same with documentation).
I've used different ticketing systems since leaving the place that used Jira/Confluence, but have been disappointed at what they were using and what else I could find out there. What have you found out there that has worked better?