The problem was discussed in the Wu Li Masters. You cannot see an atom...not its protons or electrons. It doesn't matter how strong of a microscope you have. The deduction/theory that they exist is through science. Observable phenomena of a/b/c causes x/y/z. There is no atom. Just repeatable observations.

He didn't. At that time nobody even knew that atoms had nuclei. The Rutherford experiment didn't "split" the atom. He discovered that atoms had nuclei by bouncing alpha particles of of nuclei in gold foils. Until then the model of an atom is something we now call the plum pudding model. The atom consisted of positive and negatively charge kinda smeared around the region that we now understand is the electron cloud.

It is important to realize that Rutherford didn't publish his result until 1911. Amazingly, that was only 107 years ago. It is hard to believe but true. Further, nobody realized neutrons existed until Chadwick's discover. Memory escapes me but I think that was around 1930 or so.

Mendeleev used chemistry to discover and order the elements by mass. A Freshman chemistry 101 graduate should be able to begin to understand the kinds of experiments he would have done. Chemical reactions use certain amounts. Like recipes in cooking. 2:1, 3:1, 4:3, etc. This is especially easy with reactions involving gases because all of you have to do is measure Pressure, volume and temperature (we should all remember PV=nRT). So Mendeleev inferred chemical masses this way. He also realized that if he ordered the columns and rows a certain way then properties and even missing elements could be predicted. While this work predicted many elements it also was subject to error and revision. I recall a specific example where he got the order wrong only by 1 column but still wrong. I think it was Tellurium and something else. I could be mistaken, I'm not a chemist. Still it is impressive work. It is amazing to think of all of chemistry and thought that went into his work.

The modern science of x-ray and mass spectroscopy has come into existence from a different direction. Namely these are the results of humanities struggle to understand radioactivity particle physics. I am professionally informed on x-ray spectroscopy specifically. The results from these pursuits have dramatically confirmed Mendeleev's work. Our understand of the atom, the electromagnetic forces that govern it and the underlying order is absolutely impressive. Support for the model that Mendeleev formulated is overwhelming.