Why did Rutherford choose Alpha particles for his gold foil experiment?
A better question would be "Why did Rutherford choose gold foil for his alpha experiment?"
The sine qua non was the alpha particle. It was a new probe, particles ejected from newly discovered radioactive polonium. What could he probe? Physicist/mathematicians can make conclusions if they have data, and one of the possible results they could gather was the alpha-stopping power of various materials. Bombard the object, look for penetration.
When there is no more penetration, you have a data point. One point. One point after lots of experimental time. But, if you could have an object only one atom thick, the alpha rays would all penetrate, but some that came close to a positive charge (or negative) might be deflected. There were different theories about the position of the positive and negative charge in atoms at the time, so this would be a good probe. Rather than getting one point per object, you can get a data point from each alpha particle!
An issue was getting a sheet of something as thin as one atom in thickness. Gold is one material that can be beaten so thin that you can see thru it - not one atom in thickness, but not very many. Few materials can be beaten so thin. And gold has a large atomic number, so if the protons were concentrated in one nucleus, there could be a large angular deflection as an alpha particle passed thru the thin sheet. And there were some very large angular deflections (~180 degrees) which strongly implied protons were concentrated in a nucleus.
Electrons pass thru matter like a bullet goes thru gelatin: friction with the electron cloud in atoms just slows the electrons down till they stop. Neutrons generally don't interact with matter; difficult to generate, difficult to detect. These particles would not have given such useful information, and this knowledge left alpha particles as the only interesting probe by elimination.