tl;dr: There is no legal way to do so that is faster than looking at the spells manually.
Besides D&D Beyond and physical copies of the source books, there is no legal way to acquire information about the spells in 5e (excluding SRD content). Since D&D Beyond doesn't offer what you want, and you don't want to scour the books manually, you don't really have any legal option available.
What you can do is find a website that allows you to full-text-search SRD content. I don't know such a site, however, and I believe asking for one specifically would be off-topic here. Obviously, this only covers the spells in the SRD, but it's a good start as opposed to nothing.
Since your profile indicates that you're web developer, you could search the source code of the HTML documents listing the spells.
More specifically, you could iterate over
https://www.dndbeyond.com/spells?page=X, where X is the page (duh). The links to each spell is in the format of
<a href="/spells/spellname" class="link">Spell Name</a>. Regex-searching for those links (and excluding those that link to classes, depending on your regex search term) yields a list of the HTML documents for all spells.
The spells' HTML documents contain the spell description inside a
Considering that you're a web developer, you should be able to figure out the rest yourself (I'm not sure if it would be legally ok if I provided a full implementation, even if I had the spare time).
Either way, this might not necessarily be faster than manually searching the spells, but if I had to do this, that's probably how I would do it, just because it feels efficient (even if it's not). The curse of every computer scientist ^^
Do note that, depending on how you implement this, you're going to have to make your script login to D&D Beyond, otherwise you won't be able to access non-SRD-content. Or you download all the non-SRD-HTML pages manually, but that defeats the whole purpose of automation.