Don't know if this will help but this is my DP's take on sex and intimacy. (He thinks much the same way as your partner, btw.)
He says that he's been in relationships in the past where, because they weren't really suited or working as a couple, sex was THE way of being intimate.
It was in a way like sticking their fingers in their ears and going "la la la - our relationship must be ok because look, we are having great sex and that proves we are intimate, so everything is fine . . ."
Since we've been together I think we've both learned something completely new about what intimacy is. That it can be, as you said, cooking together, catching each other's eye, even sharing a favourite film. Sex is just one way of being intimate.
Perhaps being intimate with someone is kind of like standing side by side, holding hands and coming at things from the same direction.
It's a word that has a lot of sexual connotations, but actually I think you can be intimate with friends, with family . . .
Hmm, and thinking about it, perhaps it's also the fact that when it's a healthy partnership, you are having a relationship with the person INSIDE, not the physical body.
If you are in love with the person inside first and foremost, then the body is a secondary thing - it's - oh I don't know like the colour of the car you buy. It's great if you're crazy about the colour of the car, but it's the quality of the engine that really matters.
I guess all I want to say is that it sounds to me as if you are doing just fine on the intimacy front, going by what I see intimacy as being. He sounds lovely.
Perhaps there should be a different word for intimacy, to get away from the sex thing.
So do you feel these things with him:
Closeness
Affinity
Rapport
Attachment
Warmth
Friendliness
Because that's all intimacy, isn't it?