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.
But are usually washed before you get them.
You’ll know filthy unwashed eggs when you see them.
Boiling water also kills bacteria. In fact they get killed long before the water starts boiling (pasteurization).
However, water contains salts and minerals so I always give it a quick scrub and rinse and dry off, but I’m certainly not worried about my pot being unsanitary because I boiled eggs.