I'm glad you are still working on figuring out what was going on that led to you cheating.
I think it goes deeper than this. This process is like peeling an onion and there are often a lot of layers. Whenever you come up with something, ask yourself why. "We were having problems in the relationship - WHY did I feel that cheating would fix that?" "I was feeling lonely, WHY did I cheat rather than talk to him about it?"
You said that you would feel lonely and lost without him, but is that a fair reason to stay in a relationship? Are you comfortable being alone or single?

If you choose to stay together, it will be hard, but if you truly love him, losing the relationship will also be hard. There isn't an easy way so don't make the choice of finding the easy one. Choose the one that's right for you.

I do think that you can work this out but you have to decide whether the husband-qualities he's missing are a requirement for you. Are the things he has on your list enough to make you happy? How much value do you put in the ones that he is missing? Understand that relationships (long term) and marriage isn't always easy for anyone. The media often gives us the fairy tail version of it "meet, fall in love, small problem, marriage, happily ever after". I'm not saying there isn't a happily ever after in real life, but it takes work, don't shy away from it because you'll find it with everyone you are with.

A lot of people see counselling as failure. I believe it's a huge societal problem. But it can be hugely beneficial. Would he be willing to work through relationship books instead of counselling? (I can give you some good ones if you think he'd agree)

I'm sure the email he sent was difficult for you to read, but I think it's a good sign that he is turning to you with his feelings. One of the hardest things for his situation is that the person he is used to turning to when things are tough, is the person who caused him all the pain.