It used to, I think...
This issue is on an interesting state. It will start recording profiles and stop, it will show the name of the profile, but you cannot record multiple profiles and compare between them. The current state (Safari 9 beta on El Captain) is only working as a trigger to the "timelines" tab start and stop button.
I did not had a chance to try putting this in the middle of the code and trying in the middle of a page load process, but would be interesting to find out all implications.
This API is still useful and works well on other browsers, although MDN says it's non-standard.
If it is showing the name, then this isn't a bug anymore.
The other issues you mention should be filed as separate bugs. Yes, console.profile will work in the middle of code and do a full Timeline recording for that section of code.