In the bicarbonate ion, why can the hydrogen not bond to the carbon?
First of all I appreciate your effort to think out of the box.

Now, as you are thinking of a new structure for the bicarbonate ion, you just have to compare the stability factors between the two different structures and you'll get your answer!

Here let's compare the structures:

Stability factors for the conventional structure


Octet saturation in case of each of the element.
Resonance stability (moreover, two equivalent canonical structures give more stability) in the molecule.
The negative charge resides on the more electronegative $\ce{O}$ atom.


Unstability factors for your proposed structure


Octet unsaturated in case of the positively charged $\ce{O}$ atom.
No resonance stabilisation in the molecule.
(your point) Energetically disfavoured due to numerous charge accumulation.
The positive charge resides on the more electronegative $\ce{O}$ atom.


That's why your proposed structure isn't the actual structure of bicarbonate ion.