What happens when you let your son paint without supervision!
The short answer is: you don’t get it messy in the first place
A sponge or cloth is not very good at getting the food detritus from the sieve. You wipe and just move the particles around. A running tap might work if the upturned sieve is washed through with running water but you have to ensure that the whole surface area has been douched. The bristles from a brush might be pushed through the grill on the sieve, but this is somewhat tiresome and tends to move dirt from one place to another.
Ultimately it is a lengthy job and not one that need be done. Drain the veg with the saucepan lid. Maybe use coffee filter inserts but never use a sieve. It’s just not worth the cleaning effort. That’s why you never see them in Star Trek.