Horse people (g) doing business badly. Again.
No disrespect intended, OP, I don’t think you did anything wrong.
They may not have gotten back in touch because in their mind, they erroneously thought that the agreement was in place and the paperwork was incidental and not important to complete in a timely fashion. Not realizing that you couldn’t read their minds as to price and terms, or even know their intent re the lease.
That’s not your problem, of course. But be ready to counter any fake bad news they spread about you. Just tell your side of the story and keep it simple, and let the listener draw their own conclusion (that the other party was in the wrong).
That said, it would have been more professional on your part to have followed up diligently, yourself. If they stay silent, then down to a final email from you “if I haven’t heard from you by Nov. 30th [or Dec. 15th or whenever] then I will assume that we will not have a lease in place for 2024.” It’s not a terrible thing that you didn’t do this. But keeping up communications even though they didn’t would have cleared up a lot of uncertainty.
For the horse owner, ending up on Jan. 1st without the lease they had planned, and having done nothing to acquire a new lease with someone else, they did put themselves into a financially awkward situation of taking of financial care of their horse. Assuming that they did need the lease to help them keep the horse. But that’s on them, of course.