In other words, "knowing" math meant pretty much diddily-squat unless I could formally and rigorously write out proofs for everything I thought I knew.
You appear to believe that somebody can "know" mathematics without being able to prove things, but that their mathematics knowledge is devalued by stuffy gatekeepers who insist that all mathematicians should also be able to prove things. This is nonsense, akin to the idea that a skilled fiction writer should also be able to tell stories, or a skilled athlete should also be able to compete in sport.
When a textbook says "your childhood is over" and now proofs are important in mathematics, that is like saying to a writer that it is no longer sufficient to be able to spell words and construct grammatical sentences, because while those abilities are fundamental to writing, they don't make a writer. The next foundational skill the writer needs to learn is to tell stories, and the next foundational skill a mathematician needs to learn is to prove things.
I've always been an intuitive, heuristic learner, combining past knowledge with intelligence and intuition to determine what a reasonable answer is even though I cannot "prove" it to be 100% correct.
To me, what that means is that you frequently made guesses and you weren't generally able to justify why those guesses were correct, perhaps you weren't even always aware when one of your assumptions was a guess rather than something you really "knew" to be true. This approach may well have found you great success, but only because the mathematics you were doing was "too easy": easy to get intuition for and easy to make correct guesses about. But once "your childhood is over", being a good mathematician requires more than the ability to make guesses that are usually "reasonable".
I'm sorry if that sounds harsh, but I think it is a realistic assessment of this kind of attitude towards mathematical rigour. A student with this attitude is in what Terence Tao calls the "pre-rigorous stage".
To that end, are there [...] sub-fields that could be used to inspire students interested in research-level mathematics but who are weak in terms of proofs or do not find proofs interesting enough[?]
Yes, there are surely plenty of research areas in mathematics which don't "focus" on proofs; see the other answers for examples. But "not focusing on proofs" does not mean that one can become a good researcher in those subfields without learning how to prove things; rather it means that proofs are merely not the end product of that research. Good researchers in these areas (or in all areas, I should think) are at the "post-rigorous" stage, not the "pre-rigorous" stage.
To inspire students who struggle with proofs, I think "don't worry, you can still do mathematics if you're bad at proofs" is the wrong message. I would rather say: You can learn how to write proofs, and I will help you.
Likewise, if students find proofs uninteresting before undergraduate level, I think "don't worry, you can do all the other things and not focus on proofs" is the wrong message. Rather, I would say: The things you're proving now are not that interesting, sure, but that's because these are basic exercises to help you learn the basic proof techniques. But there's lots of interesting mathematics ahead of you, which you'll need proofs for, and it's much more fun once you're proving things that you do care about.