I don't (read can't) answer a lot of questions, but I do ask some questions. Nothing makes me want to say "thank you" like a quick answer that gets me out of my jam. I feel the need to thank the person, but I know that's discouraged in the comments. But maybe for the answerer they don't care and an upvote and question acceptance is all the thanks they need.
So four psychological needs are at play here from askers and answerers: askers who feel a need to thank and those who don't, answerers who appreciate thanks and those who don't.
If thanks, are in order, but not necessarily desired to record permanently, some functionality where when you accept an answer a pop up box comes up that says "Say thanks or leave a useful comment for others" with a radio button to choose one or the other (and a 'skip' option). Thanks goes straight to the answerer and isn't recorded permanently, comment goes under the answer.
If the majority of people just want reputation, then I would suggest a slight change in wording for the comments instructions. Something like: "Avoid comments like "+1" or "thanks" (an upvote is all the thanks we need!)"