I'm in exactly the same boat as you OP, simular age, also have a grown up teen so not wanting to date me with young kids or who wants kids. Also have a successful solo life, but missing that extra bit of the puzzle that most others get. Also stay single without dating for years at a time.
Some of the replies you have got I find frustrating as they all jump to the same assumptions and predicable 'on the plus side most relationships are shit and being single is better' easy to say when you have done life with a partner. Why don't you just leave your partner then if it seems so glorious.
Wanting to partner up doesn't mean you are desperate for it. As you say, if you was desperate you would be with anyone, never single. I could get a partner this week if I would take anything. But want someone right for me with mutual attraction, shared interests, intellect and life goals. That's not so easy to find.
Like you I also got chatted up anywhere and my looks havnt declined that much at 39. There are just less single men our age about now, so most you happen to walk by in real life will either be coupled up or assume you are even if they are single. As you say, coupled up friends are busy living their own family life most times. And even when I spend the day with friends, family, they go home to their partners at night and reflect on their day, I come home to my dog. There's noone there for the spontaneous 'shall we nip out and do this' for an hour. That doesn't mean I'm desperate, I'm most certainly not. Making strides and the most out of life in every other department I can. Many things I'm focusing on and enjoying the most I can. Doesn't make that natural biological thoughts of feeling i missing out not having a partner to do life with though for that bonus on top.
I'm not giving up on online dating, as I think odds are slim to find someone who happens to be simular age, single, attractive and ticking many other boxes in real life. I'm searching all the single ones on a 35 mile radius and still rarely finding 1 I like the look & sound of. I will put more effort into it when summer is out of the way and find more time to sieve through the shit ones, then shit replies, then waste of times, then shit real life dates, arr it is a lot.
Good luck with your search too. It is hard. You are not alone. I know 2 others our ages in the exact same boat too in real life. Both also beautiful successful women. I do think that actually puts some men off as it can be intimating so have the woman to have achieved more in life. The ones that have been mega successful and will outearn most women by miles, will be buying themselves a sugar baby. Those on par will usually still want to be the higher earner. Those a tad less will the the insecure ones I mentioned. And those with much less i wouldn't want myself anyway.