I've been the "depressed and don't know what I want" spouse - and I think he's taking the piss and needs to stop yanking you around. If he actually cared about all the things he's saying he knows he's not doing - the lack of support and taking you for granted - he would do something about it, not just give lip service. To me, it looks like he's trying to make himself look like a better person by saying the words, but actually is just trying to push so you make the final shove so he's not the bad guy he is really is.
In my situation, it had been a shitty year+, family death and another getting a terminal diagnosis, lots of health problems, and I had people in my ear, in my social circle, who were essentially feeding me poison, but I didn't see that until later. The shittiness made it easier to do, but the poison was what was making me question everything from my relationship, my sexuality, even my concept of love. I was beyond messed up.
However, in trying to find those answers, I did a lot of reading, I initiated us going out more and more one-on-one time without the kids to help with the disconnected feeling I had, we talked a lot in person and over text, I sought help, and in spending more time and less with the poisonous people, it eventually clicked that - while there were some decent questions that came up (especially as we've been together since being teenagers) - a lot of it was ideological bullshit that was only hurting me for someone else's ideal. It wasn't as easy as just dropping it, but seeing it for what it was was a big step back on track.
My husband was very patient through this, put up with so much of my bullshit that I still regret even though he's very forgiving and always says it's not water under the bridge, but he was also practical about it. We were also not really in a situation to split houses at that point and he basically said 'if you really feel this way, we can rearrange the house. I'm not going to push you to leave or toss you out, but I don't want to share a bed if you don't see and love me that way". He works nights to my days so that wasn't entirely needed for us not to be literally in bed together, but he was making a clear line that I couldn't just keep talking & reading about it, that this was going to be the next consequence going down this path. Between that, his support and outside support to help me see the BS for what it was, it was probably still a good month or so before I really felt like I was back in my marriage.
While he might have people in his ear, I think your husband similarly needs a next consequence with a deadline because it is unfair to you for him to just keep talking and doing absolutely nothing else. Even if he doesn't want to see a doctor yet, he needs to be putting something in - other than his dick - to show he's interested in trying to figure this out. He's not going to get answers just in his own head - if they were there, he'd have them already - he needs to be doing something with you, even better would be you and outside support, to work through this if he actually wants to do that.
Otherwise, yeah, see someone about your financial & legal situation and figure out how to rejig the house because this isn't good for you or the kids. As someone who had parents who did this kind of thing for far too many years, it can create a lot of harmful instability and we were all happier once things were finally settled.