I am actually going through something that is in some aspects the same, yet as a whole kind of the opposite. I have close friends, but they no longer live in the same city as I do. I emigrated for work two years ago. I didn't really like the city much, and at the time I also felt kind of heartbroken to say goodbye to my friends.
As a result I too did not feel like making new friends, but for opposite reasons: I did not want to root down in this new city, and then break my heart all over again when work finished (I knew it was a temporary thing). So in the last couple of years I have spent most of my time on work, and only "mildly" socialised. Slowly and almost unnoticeably, the relative social isolation has worn me down.
Last year while travelling I met the woman that I want to share my life with. As she lives in another country, we have been in a long-distance relationship (despite both of us having bad experiences with that before and swearing we'd never repeat that). In some sense this made the social isolation worse: we spent so much time on Skyping every day that we forget to meet our friends.
My contract is nearly ending, and our plan was that I would move in with her soon. I am really looking forward to this, and to end this period of solitude. However, she has lived in her city for nearly a decade, is "settled in" and has a network of friends. By comparison, I'm moving to for her - while the city she lives in is nice, I would not have thought of moving there if not for her. I will also have to build up yet another network of friends (like you, I am in my thirties, and I too notice how this is getting harder).
This created an unbalance in our relationship: I "need" her much more than she "needs" me, as I would have noone else in that city when I arrive. This asymmetrical dependency resulted in a a strange kind of tension, making me hyper-oriented towards her to the point where it resulted in me becoming unstable.
So now we changed the plan: I'm looking for a temporary place to live, in some kind of shared appartment arrangement with (ideally) lots of roommates to socialize with and explore the city with. That way I can find my own ground in the new city and discover on my own terms whether I'd like to live there, and what I like about it. We will still see each other most of the time, and after a few months still intend to move in together. But this new plan will take the pressure off the relationship.
I am all with you that friends are often a bit of a distraction, but to me the good friends are an enhancement: they support and enable more than that they distract. I hope that some day you will find a few of those kind of supportive friends, and that they will ease your burdens. Until then I am glad to read that you have a supportive partner and family! And I wish you good luck with parenting.
(aside: after posting my message I was actually a bit worried that I would come across as one of those people who insists on constantly trying to cheer up their depressed friend, but actually only puts more pressure on them that way (I have experienced that). I'm glad it didn't come across that way - well, either that, or you're too polite to mention it)
First and foremost I have one very good friend (my partner!), and a rather good relation with my mother (my father passed away, alas). I consider my mother like a friend, who given her age needs me more and more and this will increase. Same with my mother-in-law. That's both OK, but also a burden of stress.
It appears difficult to become friends as you grow older compared to being young. I burned ships with my old friends because the relations were no longer mutual; no longer developmental ("stuck" or "grown different paths") or downright parasitic. Ironically, my last major friendship was my best friend wanting to use mushrooms with me at my place while I was no longer interested in recreational drugs. I was an avid gamer, but no longer have time to play games so all friends who I had who were into gaming I no longer have that in common with. That's a big blow in my social network and friendship list.
The gist is, I find individuals inherently boring and basically I don't want [new] friends; they're a hassle and nuisance. Friends are a distraction, just like this very forum. At best, they're interesting, like this forum. At worst, its annoying to waste a Sunday on a birthday (with mindless chit-chat, annoying opinionated people, eating crap I could better buy/cook myself, and travel time/cost), just like you can waste an Sunday on this forum. Heck, like Facebook, rather. OTOH, you reap what you sow. I still got like 100 movies to watch if not more, 20 books to read, etc.
My family and partner's friends and our newborn is more than enough. I need to focus on improving those connections first and foremost since my autism does put those relations under pressure both generally as well as incidentally more so. That plus integrating my autism diagnosis with my career which isn't easy in mid 30s. You could argue I'm not ready for it yet but I didn't start taking SSRIs to become more social. I do want to meet more people with autism, but that's out of self-interest, and a long-term project since right now I still feel overburdened due to the above (mostly our newborn).