The bottom of the harbor.

Obviously you hang them up to dry. That way you can reuse them.

American here, so probably ignorant AF, but how wet are your teabags exactly? Everyone I've ever seen make tea does exactly that: pull it out of the water, smash it with a spoon against the cup rim to squeeze water out, toss it in the trash. It's barely more than damp after that.

Are the tea bags there so very different from ours that there's enough water retained in them that it risks making your trash can soggy?

(Tone here is meant to be curious, not confrontational; honestly wondering!)

It's biodegradable, right? Straight to the compost (or wherever plant leftovers go).

Put them on the metal bit on the side of the sink (not in it), then later once it's cooled put it in the bin or food waste bin depending on if the bags are biodegradable. No smell, and the sink remains fully in service.

Soap dish or similar.