You dumped out the water ten minutes ago. The pot is cooled off, and empty. It only had whole eggs in it, still in the shells. Now tell me, do you wash that pot? Get out the sponge and the soap and scrub it like usual? Or do you give it a quick rinse and slam it in the drying rack? No judgement here, only curiosity.
Depends on the pot… I have a bunch of cast-iron & carbon-steel pans and most of those would get a hot rinse, and then a wipe with an oily cloth after going back on the element to heat back up & dry off any excess water.
My cheaper saucepans I probably wouldn’t bother unless there’d been a mishap and an egg had cracked - but we also have really hard water here so its often a good idea to wash from time to time.