Doesn't the neutron's lack of an electric dipole moment conflict with the concept of baryons having a radius?

The neutron does have a nonzero magnetic dipole moment which was detected almost 100 years ago and taken as direct evidence for it possessing internal structure, which was much later identified as consisting of quarks in communication via gluon exchange.

The deep inelastic scattering experiments that revealed the presence of quarks allowed us to determine the structure function for the proton and the neutron, which indicated that the quarks inside them were not coincident and also that the bulk of their momenta were carried by the gluons.

I recommend you read Riordan's The Hunting Of The Quark for a readable exposition of exactly how all this was worked out.

[...] he only logical explanation would be that all three quarks are point particles at the exact same point in space.

Classically, the electric dipole of three point charges $+2q, -q,$ and $-q$ at respective locations $mathbf x_1,mathbf x_2,$ and $mathbf x_3$ is

$$mathbf p = sum q_i mathbf x_i = q[2mathbf x_1 - (mathbf x_2+mathbf x_3)]$$

A vanishing electric dipole moment therefore corresponds to the condition that $mathbf x_1 = frac{mathbf x_2+mathbf x_3}{2}$ - that is, that the $+2q$ charge is in between the negative charges. It does not require the three particles to occupy precisely the same location in space.

The generalization to quantum mechanics is a bit more subtle, but the same idea holds. A vanishing electric dipole moment does not imply that all of the constituent bits of the charge distribution are in the same place.