I always thought a delay was for the loop.
That's the theory, yes. Or even better, leave it to the mixer board.
Surely, a delay in front of the amp with overdrive would give it a very weird tone
It won't obviously yield the same result as the other way round, but depending on the exact setup and amp, you can got some nice things too.
FWIW, I currently use a "dirt boost" OD => delay setup in front of my 6V6S-modded HRDx (drive channel with low/mid gain settings - say about 3.5 / 12 - and power amp starting to breakup), so it's kind of a "distort the delayed distorted signal" thing. Works better for me than having the delay in the loop (specially with the very crude HRDx loop), the delayed signal blends better with the direct one and there are some very interesting "strange harmonics" thingies happening.