The results of _all_ of these experiments do not fit with the naive theory. And they each can be fit perfectly by adding in dark matter. But that's not important; the crucial point is that all of them can be _simultaneously_ fit perfectly by adding in the _same_ amount of dark matter. That's practically the definition of what a good scientific theory should do. Put in a single parameter and explain a hundred observed results.
Meanwhile, modified gravity theories have fared extremely poorly -- they were originally designed specifically to fit galaxy rotation curves, and accordingly it is very difficult to massage them into fitting anything else. Usually if you get the rotation curves right, the astrophysics and cosmology come out disastrously wrong. You could probably get it right if you added a pile of ad hoc parameters, but that would be bad science.
Unfortunately every discussion of this subject ever just hyperfocuses on the nearly 100 year old galaxy rotation curve observations... probably because it's easy to understand.
> Is there any prediction that would prove the existence of dark matter?
If dark matter interacts in a non-gravitational way, and is present near the Earth, then the smoking gun would be directly detecting it in an terrestrial experiment. (People also work on indirect detection, by looking at possible products of its decay or annihilation elsewhere in the galaxy, but this is less definite because such products could be made by something else.) Of course in all these cases, the specific kinds of predictions depend on what you think dark matter is made of. Unfortunately, its good scientific properties (i.e. fitting a lot of data with remarkably little input) also mean that we have very little to go on here. Practically any kind of new "stuff" that interacts weakly electromagnetically could work.