Why should I spend any of my time or energy on an unproven claim?

Probably for the same reason we dedicate energy and time to world matters of import. It's interesting, it's potentially very important, and even if we personally may not have a hand in the verification or outcome, this matter ultimately affects us all. It could literally change the way all humans think about intelligent life and the possibilities for humanity in the future.

Should I also entertain the idea that an invisible pink teapot is orbiting earth until it's proven false?

Your Russel's teapot is pink? Neat.

What if there is someone with "credibility" claiming they have (classified) photos of the teapot, is it worth considering then?

Yes! I mean the analogy is rather breaking down here because the implications of a pink teapot in orbit around the earth today would probably point to someone launching it into orbit, and not something as consequential as the verification of non-human intelligent life, but yeah, we should investigate credible claims of things that matter in general.