You're almost 40 and using "she doesn't like to go hiking" as an excuse to break up with someone. That's fine, you're not that into her, but don't say "she's not outdoorsy enough" because frankly, it's cringey.

When I was younger, I'd try to rationalize why I didn't like a guy who seemed decent/good. I would just make up excuses, anything, to not feel guilty for not being into him. It's okay to not be into someone, but the ethical thing to do is to realize that asap and not lead people on or start things you know will end badly for them. I'm not saying you did it on purpose, but you need to be less messy before dating someone new. Take some time to think about your values, goals, wants, needs, be more aware of your emotions, work on self-awareness in general.

You've already ended things with her. Asking for time/space after dating for just 2-3 months effectively ends the connection. It's also cringey and even a little manipulative to be all "I'm not ready to give up on us yet". Dating for a couple of months doesn't make you an "us". I would personally interpret this as you ending things and would be surprised and unimpressed to hear from you again.