The single op-amp IC in a DoD 250 Overdrive, will eventually distort and contribute to the distortion sound, but only say one third. However, the 250 is basically a distortion, which gets most off its grit and compression from hard-clipping asymmetrical diodes and its character from the C3 + R7 + A500K gain-pot EQ, which basically gets brighter as the gain gets increased.
IMHO, that is where the IC deployed is important. The LM741 is inefficient and smears the sound, which tames the highs from being harsh. Chips with faster slew rates or which are FET based, will make the top end harsh. That can be tamed by changing the C5 22pF bypass cap between pins #2 & #6 to a higher value.
Additionally, the pro use of a 250 isn't often discussed. Its forte is pushing a thick amp set just past edge of break-up, into a brighter crunchier sound. IMHO, best on British-style amps and disappointing on Fender amps which are not cranked with the treble settings on low.