Confusions about inductive effect and acid strength
The result is correct but the explanation is bad.

The main reason is that when you compare CH3COOH and CH3ClCOOH , the main difference is that the alpha carbon in the former has more electron density than the latter one because of which when the conjugate base gets formed, it is more stabilised than ethanoic acid because of low electron density. 

You are correct about this "In my mind, to borrow some charge density you have to be more electronegative than what you're attached to, and the more deficient carbon should have a greater pull".

Edit 1 - The orbitals of CH3COOH can provide more electron density than CH2ClCOOH due to the former having more electron rich orbitals which interferes constructively with the orbitals of other C to satisfy its demands. Think in terms of how Hyper conjugation or resonance works in terms of interference of wave functions.