You don’t at all come across as ‘untethered or flaky’. You do, however, sound like someone who has very specific ideas about what the ‘right’ friendship looks like — ‘deep’ and involving you taking it in turns to ‘take care of each other in the trenches”, but not too time-consuming at the foundation stage, and in any case you say you don’t have time to ‘grow and take care of’ a new friendship now.
Obviously you’re not wrong to want what you want, but think these beliefs are limiting you.
For a start, that’s a lot to put on one friendship, especially one you’re growing from scratch. In my experience, you get support from different friends and acquaintances at different times, which means there’s always someone, and you’re not putting undue pressure on one friend to be your sole ‘ride or die’ person. For another thing, in order to get one or two friends who will (eventually) be there for you through everything, it’s a numbers game to begin with, like dating. You need to expose yourself to a lot of acquaintances, some of whom may become friends, some of whom may become very close longterm friends. Putting all your eggs in one friendship basket isn’t a good idea, especially at the outset.
And bluntly, good friendships take time to develop. You say you’re exhausted from young children and haven’t the energy now to nurture a friendship, which I get, but I think you should try to think of something you will genuinely enjoy for its own sake, and that will expose you regularly to a bunch of different people. And you should do that now, once a week as a minimum, as a break from your kids. That way, by the time you have the energy to invest in ‘growing’ a friendship, you may have some potential candidates in mind.
I’m a sociable introvert, whose friendships are important to me, but I’ve never got, or tried to get, to the stage where I can ‘recharge’ with even my closest friends. Friends for me are for stimulation. I recharge alone, and o need a lot of alone time. That hasn’t stopped me having close, longtime friendships.
Though I should say that life hasn't helpfully put one of us in the trenches and one of us untrammelled and able to help, at appropriate times. One of my closest friends was hospitalised for a severe eating disorder during Covid, and I got a cancer diagnosis literally while I was on my way (three hour train journey) to sit under her window which was the closest could get to visiting. I needed to seek support from a different friend in that scenario. My hospitalised friend hadn’t got the bandwidth.
TL; DR. Don’t restrict yourself too much.