The $\alpha$ particle, attracted by the electrons on the outer shell of the pudding, orbits nearly parabolically around the atom, causing the near-180 degree deflection angle seen.
This wouldn't happen because of momentum conservation. It was reasonably established in 1909 (when the gold foil experiment was done) that electrons were light, so if an alpha particle were to be reflected by interaction with an electron, the electron would be kicked out of the gold atom with even higher velocity in the opposite direction. $v_e \approx \frac{m_\alpha}{m_e}\Delta v_\alpha$
Besides, in the plum pudding model, the electrons are distributed throughout the atom, not all on the surface. The same reasoning applies, though, regardless of how the electrons are distributed.
A gold nucleus is much more massive than an alpha particle, so it can reflect the alpha particle without recoiling very strongly itself.
The $\alpha$ particle hits a plum pudding nucleus directly, and because the nucleus consists largely of positive charge, it is deflected by nearly 180 degrees.
The defining feature of the plum pudding model is that there was no nucleus of positive charge. What you're describing here is exactly what did happen, it just wasn't a prediction of the plum pudding model.
The usual derivation of the differential scattering cross section makes the assumption that the mass of the target nucleus is much greater than that of the incoming alpha particle. This is saying that the nucleus does not recoil when it interacts with alpha particle.
Better still assume that the $\csc^4$ formula derivation is done in the centre of mass frame of the nucleus and alpha.
Since observations are made in the laboratory frame a correction has to be applied to the $\csc^4$ formula and that correction does depend on the mass of the nucleus in relation to the mass of the alpha and so changing the number of neutrons whilst keeping the number of protons the same does change the angle of deflection.
The smaller the nucleus the greater the correction term when the number of neutrons changes.
There are many Internet sites and textbooks which derive the correction term and here is one with the centre of mass derivation shown on the previous page.
Best Answer
Rutherford's experiment looked much like this:
(Image source)
As you can see, the incoming alpha particles hit the gold foil and could scatter in multiple directions, but the detector went around the whole foil (sparing some small region so that the alpha particles could enter the experiment) so even back scattered particles would be detected.