How should a high-school student approach professors in research institutions about research opportunities?

It might be better, if you can arrange it, to have one of your teachers help you with this. It may well be that one of your STEM teachers went to one of these universities, or has some contact there. If so, they can help you make contact, vouch for your seriousness, and ease the way.

This is much harder now in the pandemic, but perhaps it can be arranged in virtual space. A phone call from a teacher to a professor before you make contact will make all the difference. Even if none of your teachers have any contacts, a mail from them, introducing an "interesting student" will make your contact much more likely to bear fruit. Make sure that your teacher can say some interesting things about you and your interests and skills.

This should avoid the "bulk" problem and you can approach it one step at a time.

Write an email to each professor like this:

Subject: High school student looking to intern in your lab.

Body:

Dear Prof .....

My name is ..... I am a highschool student at ......

I am really interested in your work on (Clear specific details about the work taken from their college website)

I have read your papers ...... (Pick one or two open access papers that you can download for free).

I have the following questions .... (Ask some good questions)

I would really like to do a short intern project in your lab working on .... (topic from papers) possibly doing experiments ...... (Suggest some experiments similar to the ones in the papers).

Thank you for your time, and I hope to hear back from you.

Best Regards

.....

It's a high bar, but it's your best shot. Remember quality is better than quantity. Five Good emails like this will do better than 100 boilerplate emails.

Also don't get disheartened if the professors don't reply, they are very busy and probably didn't open your email, or possibly don't have the space to take on a high school student.

Also remember that it's not required for high school students to have work experience in labs. You have to demonstrate that youre really good to get an intern project as a high school student. Most people who go on to do STEM degrees and PhDs didn't get work experience in a lab as a high school student.

I guess you should not ask everyone simultaneously. Ask sequentially/serially, waiting for a response (or, after some waiting period, a non-response) from one before you email another.

As you already speculate, you should absolutely customize your communication with each person. After all, not such a huge cost for what you hope! :)

I should say that I myself tend not to respond positively to emails that are apparently boilerplate, or bulk emails (using bcc), or ...

We can all understand the apparent economy of "bulk emails", but it will also, correctly, be perceived as spam. So, self-defeating if/when you are dealing with serious people.