Hello Petra5ed: I wanted to add a couple of thoughts here. They may, or may not be relevant. First, I just want to say that I get the impression there's allot more going on here than simply a lack of emotional connection & physical contact. Perhaps I'm wrong.
I don't know how old the two of you are or how long you've been married. My wife & I have been married for over 30 years. I guess you could say we also don't have allot of emotional connection or physical contact anymore either. But that's just the way things tend to go as couples age. My impression is that many couples our age, assuming they're still together, are sleeping in separate beds, if not in separate rooms. But, perhaps you two are still young.
I do think that what you're describing is not necessarily unusual for men in particular. The older we get (relatively speaking) the more withdrawn we tend to become. (And I think this starts fairly early. As I recall males supposedly reach their sexual peak sometime in their teens.) This is, I think, part of the reason all of the men, here in Minnesota, put icehouses out on the frozen lakes during the winter & go ice fishing!

Also, of course, depression can play a part in this. And it's possible your husband is struggling with his own depression. This would surely add to the natural tendency to withdraw I think men have.
I'm also not particularly surprised by the fact that your husband says he'll do something but then does not do it, or does the opposite. Men not listening to what their wives are saying to them is kind-of a standing joke. But it's based on reality. I probably hear about half of what my wife says to me (on a good day!) Otherwise, my mind is just off in a fog somewhere. I try to be as attentive as possible. But there's only so much I can do to remedy what seems to come naturally.
So, I'm not suggesting anything with regard to what you should do about your relationship concerns. I just wanted to offer my perspective as a long-time married older male. My impression, from reading your post, is that the concerns you have may not necessarily have so much to do with your husband in particular. They may be more related to the failings of the majority of men in general.

My best wishes to you both.