I get how partially burnt TP in a fire ring could cause some to turn up their nose. But I'm thinking the only repercussions with thoroughly burnt TP is only in the mind.

I like a sparkling clean site and it is pretty common for me to make an actual fire every evening (and sometimes in the morning before I leave) to consume any garbage and TP. I would not advocate trying to completely burn TP with just a flick of a bic, not happening IMO most of the time.