The smallest I found was the campingaz party grill, powered by the cartridges
This cooks fine, it was cheap, the gas is cheap, readily available, non stick so easy to clean, comes in a small pouch, easy to light, you can use it on top of your external table without fear of damaging anything or covering it with ash or setting fire to stuff
The cooking area is sufficient for 4 normal adults, or two greedy ones.
I would not worry about storing the cartridges. You might get a week to two weeks of daily BBQ out of one, and they are very clean, compact and easy to store. We find it helpful to have indoor and outdoor cooking options at the same time - and I can't be arsed lugging the main bottle outside.
This also gives you a backup burner (this thing is designed to be used as a regular gas burner too, you just take the grill off) so you can be more relaxed about the main bottle running out when camping.
It is not the full charcoal experience, but it is the right balance of taste v storage v cleaning in my opinion. I can't be arsed messing with the main gas bottle when I want to cook outside. I can't be arsed with ashes or storing charcoal either. And I really can't be arsed with items that are difficult to clean