An interesting range of experiences on orbits. I have been asked to orbit on downwind, at the downwind to base turn and before entering the ATZ - but never on final.
I have had my share of tight approaches where the guy in front isn't going to clear - other than making sure I am at 1.3Vso making gentle S turns and being mentally geared for a go around the whole way down - I watch until it is clear the gap isn't going to happen and then am on the go. I have never thought about doing a 360 in something like landing configuration at say 400 ft and would reject the request in favour of a go around any day. A few minutes wasted but able to stay right in the mental mindset for landing (i.e. assuming you are not).
Possibly influenced by my first solo land away - I was cleared to land after the departing twin - who promptly spit clouds of black smoke and engine bits! My instructors words prior to me going off were "if there is anything you don't like about the landing then - full power, pitch up, clean up and do the pattern again" and that is exactly what I did.