In 2005, it if had a HETAS cert it should have a flue, so assuming it does, a CO monitor is a tenner. I don’t know about an air vent, but I can’t see it being thousands. (IIRC you don’t need one if you have trickle vents in the windows, but I could be wrong). Chimney should be swept annually, so that’s neither here nor there. Just do it.
You definitely need a flue (my mum didn’t have one pre-HETAs and her chimney caught fire!)
Bear in mind the surveyor may know very little about chimneys. Mine flagged up as cause for concern here, and the surveyor had not realised that it was in fact merely decorative, and therefore could not be unsafe - You’d think they’d spot that!
You have several options.
Do the work and you pay for it
Reply to the buyer and tell them that of course you want them to have a safe property and you will be removing the woodburner forthwith (likely more expensive than a vent, I reckon, but it would concentrate their minds while leaving you looking good)
Assume this is an opener for negotiating on price and tell them firmly that you are selling them the house as is, for the price agreed and if they want to do work on faffing about with the woodburner they can pay for it. It’s not unsafe if they don’t light the thing and they can get it done when they move in. If the mortgage company is insisting on it being completed (possible, although unlikely) you have no problem with them paying for it to be done pre-completion. It’s not like a leaking roof, it’s a completely optimal extra!
Depending on the strength of the market in your area, I would go for option b. That will likely be enough to gauge if they want it or not, although I suppose they might call your bluff. I suspect not, though. I’m about to buy a woodburner, and the guys that fit them are run off their feet!