Swimming instructor

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Hello all - I need advice. I'm 49, a divorced single mum of a wonderful DS, 9. DS has autism. Ex-DH and I are on civilized terms; I have a FT job that I love and good friends.
Last autumn I enrolled DS in a swimming group and myself in 1-2-1 swimming lessons, and we have the same instructor. I have my lesson when DS is with his dad. The instructor is basically the best-looking man I have ever seen. He is really nice, in his early to mid-30s, easy-going, and good at his job. From my first lesson, we started to have chats, initiated by him, after the lesson. First about DS and autism, then about swimming, and then increasingly longer conversations about TV programmes, families, sport, work, likes and dislikes. We've been exchanging texts and e-mails.
It got to the point where (months ago now) I had gathered enough courage to e-mail asking if he wanted to go for a coffee. He never replied. I took this as a clear signal and was prepared to scale down the chatting etc. but he has kept messaging and talking. Last week we came out of the pool at the same time after class, and he took my bus for a few stops even if he was going in a different direction, just to keep talking. Before that we had a very long chat which he started by asking me if I believed in love, and continued with him saying at one point that there are mums (plural) he fancies but they're off limits because he teaches their children. I'm not sure whether that was a hint. He has said other times that he doesn't have a girlfriend, but that he would like to get married and have kids.
So now me and DS are off for the holidays. I've again gone out on a limb and sent him a goodbye message saying I'll miss him. No reply.
I really like the guy. I'm looking for fun/romance, not a stable relationship, which would be unfeasible anyway given the age gap and what we want from life.
There's a lot I don't know about him - relationship history (can someone so handsome really be single?), why he never replied to that e-mail, and above all whether he likes me, other than talking to me after our swimming lesson! He doesn't come across as a player, ifswim. If anything, he's a bit goofy/insecure, but that could be how I see him from the perspective of age and experience.
In sum, what should I do? Change swimming pool and forget about it? Embrace my cougarness and flirt even more openly? Just continue as it is - or maybe I should ask him if he has a (much older) brother?

He's just not into you. Move on. He is just doing his job with the chit chat.

It could be he just enjoys the flirting and the attention, does it with other women too and has no interest in actually going out with you. Or maybe he just enjoys the conversations and you're reading waaay too much into his behaviour but since you're a client he can't come out and say he's not interested in you (hence why he's ignoring the messages).
I would continue with the swimming lessons but not respond to the personal messages.
I get why it was a lot of fun flirting with him though

well there is no point in any of it as he wants to settle down and have kids, while you want different things due to age gap. If he wanted anything more fun/casual, he'd gone for it ages ago - he'd jump at your coffee suggestion! I think he is a bit weird, all the personal texts and questions of love when he isn't going to date you.
I agree, stop replying to texts/emails. If he asks why, or asks you out then just be direct that you are after different things. Maybe he wants some older sister type figure to confide in - then let jim just be honest and say he only wants friendship, then see if that's what you may want.

maybe also he is keen on one mum his age and wants to sound you out whether it's ok to ask her out while teaching her kid.

Thanks for your answers - it's pretty much what I had been thinking but I needed some external confirmation.

I agree with Dynomite

I think he was just being flirtatious with no real intention of taking things forward hence why he didn't respond to your more direct messages. Tbh, you've given him the green light twice now: once the coffee invite and then the "i'll miss you" going away message. As neither has yielded a response, i.e. not even an acknowledgement it's time to knock this on the head. By all means continue with swimming lessons especially so if ds is enjoying them but time for you to widen the search pool. i love bad puns

Good luck!

I know they say age is just a number, but if he is in his early 30's, the age gap is probably too much for him.

I think he's interested but something is holding him back.

Does he know you're divorced?

Is he a bit shy about the age gap?

I think you need to get him on his own and see what happens.

I think he has already given you his reason for why nothing is going to happen. Mothers of pupils are off limits. That is why he isn't responding to openly flirty messages. It sounds like he likes you and you can be friendly and chatty together, but if I were you I would stop the romantic daydreams and treat him in a friendly but slightly more distant manner.