I have lots of old books that aren’t very good. That I can’t give away. What do I do with them? I have this strange taboo feeling about destroying books, even bad ones!
That is what those book corners are for. In Germany a lot of old public phone cells have been repurposed for that. Just don't overload them and also consider taking some books that interest you. Somehow it regulates and I guess someone also throws away stuff from time to time. I actually would not consider many of those listed in the OP 'libraries' because I rarely see books returning. There are more books exchanges. I liked the idea of bookcrossing.com that gives more value to the books (I never got any ping back though). There are also book resellers (some buy per weight, some let you estimate price by ISBN scanning before sending them)
They're worthless, sadly. You can make a free library-type thing, but nobody wants them.
The best use is to glue the pages together (one set of about 20 with the cover, one set of 20 with the back cover, and then the rest). Put it on a bench/vice and use a jigsaw to cut open the center portion, and then glue the back section to the hole you now have, and now you have a book you can hide something in.
You can try to find out or estimate how many copies there were. If there are tens of thousands, it may make you feel less bad about tossing them. If people want them, they can find them and it's not necessarily your responsibility. For many books the world's need for them really diminished in relation to what has been mass-printed at one time: popular fiction like airport books, self-help type etc.
Second hand bookstores often buy. You can offer your books to them. Some they won't want, but might accept as a donation along with those that have monetary value to them. In reality, they have similar constraints of space and expected time to when someone will purchase (which could be years for even semi-known books). People have this romantic view about books, but if you want to hold any kind of personal library and don't have a freaking villa, you have to learn to be pragmatic. Even if you don't have a lot of books, get ones that you would read.
Yeah, I wish that libraries would have the funding for a "last copy" system where at least one copy of every book which was ever in their system was kept in storage --- yes, there are systems such as the Library of Congress to serve as book repositories, but they don't loan, and driving there for research is a pain.
Donate them to Better World Books or some similar group?
If you hate them, that's all the more reason to do a destructive scan on them and upload them somewhere. If you don't care about the book, you can make a beautiful copy, and somebody wants to read it. Even if it's just to research religious fanatics around the Boulder, CO area who were in a business conflict with someone from the Jones family. They might be interesting because of who published them, because one has a list in an appendix of the businesses in downtown BFE from 1982, or because the grandson of the author became president of Chile.
I want to say that you don't know what books are worth reading until after at least 1000 years, but actually all books are worth reading after 1000 years. They become tsukumogami, somehow old and wise even if they started dumb. The list of the garbage from the 16th and 17th century I've spent all day reading that is exactly their equivalent of the self-published AI slop novel is embarrassingly long.
If there are already good pirated copies of it, cut the spines anyway and you can use the pages for fill when you're packing things. Lots of O'Reilly in my life went that way.
edit: the problem is having access to a indy print shop that will scan them for you (it only takes a few minutes, but the machine is huge and expensive.) Most of them are closed down and have been replaced with chains. I was spoiled for a while working with one.
I believe libraries should take all books. Then they can turn them into confetti if they want. In practice, of course, they are not interested at all. If they did they might find some interesting gems when someone passes away or similar to make up for it.
Just toss them. Make a real assessment on their quality or worth and if you think it’s very unlikely someone in your immediate community would not only pick it up but actually read it, toss them.