I did not understand the last sentence. If I solve the captchas, get the search results and then do the cleanup, it'll just continuously ask for the captchas and that's the only added pain, no? Will they be able to conclude if all the requests are coming from the same user?
Yes. But its going to keep asking you the captchas till the user changes behavior (sample of 1) :) (and of course the ip address too like the other poster mentioned - you can try it by searching for insurance/anything with good money on your phone and switch to desktop - assuming both are connected through the same router).
I once had to solve thousands of captchas for an archiving project, and buster helped me with a quarter of the captchas (https://github.com/dessant/buster)