Bond order of H2O and NH3 using group theory to construct MOs
It depends on your definition of bond order. If you are using the simple formula 

$$\text{Bond order} = \frac{\text{Number of bonding electrons} - \text{Number of antibonding electrons}}{2}$$

then, yes, it won't be able to capture the subtleties that you describe. After all, when you use that, you're categorising electrons into bonding vs antibonding in a black and white manner. So it's really no surprise that it doesn't reflect "weakly bonding" MOs.

You gave the example of water. Well, for that weakly bonding MO, you need to make a choice: is that considered a bonding MO, leading to a bond order of 1.5 per O–H bond in water (doesn't make much sense)? Or is it considered a nonbonding MO, leading to a bond order of 1 (makes more sense but then you have this problem you brought up)? Or is it somewhere in between, leading to a bond order between 1 and 1.5 – and crucially, if this is the route you want to go down, how do you quantify the bonding character of the MO?

I don't have any experience using different definitions of bond order, but if you used one of these, I would not be surprised to see an O–H bond order that deviates ever so slightly from 1.