Is it disrespectful to write very similar emails to multiple people? I need to write to several academics with a very similar request. Would it be disrespectful if I just copy-paste most of the body of the email, and change just the person-specific details?

On one hand, nobody likes mass-produced emails. It could be irritating for them and embarassing for me if one of them forwards the email to another, and the similarities become clear. On the other hand, what I need to say in each case is very similar, so once I write one "optimal" version of the message, it's hard to see how and why to produce another version.

For context: I have been working at a problem which is somewhat outside of expertise of me and of my supervisor (I'm a PhD student). I think I have a solution, but I am not sure how original it is and how interesting it is. It could well be that it's been known for decades, but I can't find it because of differences in terminology. Hence, I would like to write to some people who have more expertise in the field. The email explains roughly what the problem is, how I came to consider it in the first place, how the solution works, and what what specific pieces of the work of people I write to make me think they might be interested in problems like that (this is the person-specific part). If relevant, the field is pure mathematics
It is not disrespectuful per se to send bulk e-mails. Academics get them all the time: conference announcements, field-specific mailing lists, automated e-mails from submission systems. If it is appropriate for what you have to do, there's nothing wrong with it.

However, I consider disrespectful the way you want to use them. If you just want to know if something is already well-known in the field, start by contacting one researcher. Writing to 100 of them means unnecessarily wasting their time.

Also, it seems like you want to put the introduction of your paper in an e-mail. Don't do that. Keep it short. Your paper should already contain an introduction with motivation and previous research (to the best of your knowledge).

Similarly, for the same reason of not wasting people's time, you want to make sure that your paper is readable to people working in that research area. Make sure that you know the basics of the field and use accepted notation. If your paper looks like it's written by a crank, most recipients will file it directly into the recycle bin.