But is having gone through years of analysis or any sort of therapy a reliable indicator of the therapist's mental and emotional health AND their ability to help someone else with theirs?

Just playing devil's advocate. I don't see having gone through therapy as necessarily a good thing at all. There are many stories of therapy causing significant harm, making people worse. A therapist in training, like the rest of us, could easily be made worse (or just not any better) by bad therapy. Or years of training and analysis might distort their way of relating to others, with excessive focus on theoretical and clinical way of thinking.