I suppose my two cents for anybody seriously interested in independent research is to make it as easy as possible for credentialed researchers to interact with you. Learn what you can, read accepted literature, acquire some credentials yourself if possible, learn the language on Stackexchange or Mathoverflow. Definitely do not open your interactions with grandiose claims. Most researchers actually like their subjects; if you have something useful to tell them, they want to know. You just gotta make it as easy as possible for them to see you’re not a crank.

That jibes with my experience for sure. The sample size is small, but I've found "traditional" academics to be quite receptive to engaging with me when approached the right way.

For example, I cold-emailed one of the two authors of a book I was reading, and asked for some datasets they had used in their research. The professor in question quickly replied with the data, a pre-print of a forthcoming paper, and an invite to keep him informed on how things were going.

I later emailed him with a technical question about something from the book, and he replied that he didn't know the answer offhand, as the research in question was done 20 some odd years ago, so he cc'd the other co-author and solicited his response. It took a few weeks (turns out he was on vacation) but he also responded, helped address my question, and asked some questions about what I was working on.

One of the two professors in question said during some of our email exchanges that they believe there's a real chance to come up with something useful going down the path I'm going down, which is encouraging.

Anyway, I suspect part of the key to engaging with academics is:

1. (as you said) don't open with some grandiose claim.

Rather, ask a question, preferably a "good" question (that is, something that shows you have done your homework, something that is specific enough to answer, etc.). Not "Can you tell me how to get started in X" where you could have just googled those exact same words and gotten an answer.

2. Do your homework and make it clear that you've your homework. If you start off with "I was reading your paper on X, and on page 3, in the equation for flibblegazz, I don't quite understand the notation you use..." or something, it shows that you're not just pulling crap out of the air. And if you're emailing the person who wrote the paper, you're appealing to their ego a bit in that you're telling them that you care enough about them, and their paper, to dig in. I suspect that unless someone is a "celebrity researcher" to the point that they get more requests than they can handle, they'll probably be inclined to try and help.

3. Do your homework in the sense of reading the basic texts in your area of interest, read papers in the area, learn the relevant math (mathoverflow and various sub-reddits are good places to ask math questions, BTW), etc.

4. It's related to a previous point, but showing interest in the work someone else has done is usually a good icebreaker. Asking for a preprint of a paper, or asking for a dataset, saying you want to try and replicate their work, things like that.

Do those things, go slowly, and just plain "be nice", and I think you can build a relationship over time, with the folks in "proper" academia. And that could be mutually beneficial. For example, you may eventually want to publish a paper in a journal, and your academic friends can probably help you there. You may even wind up collaborating to a level that justifies a co-authorship scenario, who knows?