I don’t ship them romantically or sexually.
You can read about me rambling how and why I ship Destiel here:
Basically soulmates. Kindred spirits. But I just don’t see them being a romantic couple, or having sex. Just doesn’t fit their characters.
I agree with all of this. I’ve never seen them doing ‘couple stuff’ because as you said it just doesn’t suit. I think when they give into the love they share they will both be very happy.