continue playing unrated

I won a game on time. A position that I thought was quite interesting.

So my suggestion was to add a button in the game called "Request playing from this position unrated".

the other person could then decide if they want to play on from the position  unrated.

Sounds like an ok idea, but also good would be play this position with the computer

With correspondence chess you can choose not to win games on time automatically by checking a box in your preferences page. That way you can give your opponent as long as you choose to return to an interesting position. You can then claim a win manually if you like.

Though I suspect that the vast majority of games lost on time were also about to be lost on position and/or material.

Yeah, I agree. We really need to be able to set up all kinds of positions to play unrated.

This would be really useful for practicing positions (esp endings) against human players.