A Feynman diagram is a shorthand for a calculation. There are two different ideas that are getting entangled: is the calculation done correctly, and are the calculation's approximations (and since this is a truncated perturbation series, there will always be approximations) valid?
For Mott scattering, when calculating using Feynman diagrams the target (in this case, the proton's) charge appears only as Z2. So at 1st order, electron-proton and positron-proton scattering is the same. There's a very nice QM treatment of this in Schiff (p.141) where he shows the difference between attraction and repulsion is proportional to 1/v - i.e. for one to distinguish attraction from repulsion, the projectile needs to spend a lot of time near the target. This agrees with our classical intuition, and it corresponds to the case in the Feynman perturbation series of many photon exchanges. When texts make the usual Mott approximation that m_e is zero, that of course is far away from the large 1/v situation where attraction and repulsion are distinguishable.
For the proton's form factor, of course it has one, and in certain kinematic regions - large momentum transfer - it is important. In others, it is not: in particular, so long as the projectile wavelength is large compared to the target radius, the target appears more or less pointlike. In this kinematic range, there is nothing wrong with treating the proton as a point. (For a purely electric interaction - there's a subtlety with the gyromagnetic ratio for purely magnetic scattering) This is no worse than treating the proton as a point when applying Coloumb's Law at large distances - indeed, it is the exact same approximation.