After having raised 3 of my own boys, I can admit I see your husband's point, a bit. My first question is, do you intend to have more children? If so, allowing your son to rely so heavily on you could backfire and make your adjustment to a new baby a REALLY difficult time. If he will be an only child and will always have your undivided attention, its not as big of a deal.
My husband and I have found we each have very specific "roles" in raising our boys. Mine is definitely to nuture and teach them how to do the same, and one of his roles is to make them "tough." Now, I cannot negate his role by babying or un-doing what he does to help them learn to be men.
We believe our society is truly "raising a bunch of women" now-adays. Boys are rarely allowed to exhibit traditional boy behaviors. They can't wrestle, fight, or run and be obnoxious, as is their nature. We expect boys to act like girls and sit in a circle in school (even pre-school), use their words, and squelch all aggressive behaviors. So, how do we raise strong leaders who know how to deal with conflict and resolution (in time, the physical behaviors are replaced with strong discussions with a true winner)?
My point being, let your husband take his role, but keep up your role of the nurturer. That is a trait many of our fathers weren't shown by example, and is lacking in that generation. Some of our husbands were shown it is OK to love and be tender, and they tend to stay married over their counterparts who weren't. Help your husband know just because you have a "role" doesn't mean he can't develop his own role.
And, this clingy behavior of your son may truly be a phase he will pass through, and before you know it, he'll be "daddy's boy" and you'll feel a bit left out. Phases come and go, but how we deal with them will truly shape the development of our little men!