Electric charge attraction, unlike magnet attraction, can be neutralized, right?

It seems to me that you have a huge confusion. You should see other questions on the site, such as this one to see if they help. If they don't please continue.

If you take a proton, and put it in a room with electrons it will interact with all of them according to Coloumb's law.

You seem to be thinking in "shadowing" in which 2 charges of different sign will appear to a far observer as having no charge at all. This does not mean that the proton and the electron have each "lost" their ability to influence other charged particles, but when ou are far enough you will not notice the individual contributions and get a "canceled" effect.

Now in the "Proton-Electron" analysis, we have additional complications, because we are dealing with Quantum Mechanics, which can complicate things a bit. The electron clasically is just a particle, but in quantum mechanics an electron bound to a proton behaves "as if was an electron shell around the nucleous" Think of the electron moving constantly at very high speed so that when observed you can only see the "average" which is roughly a shell, therefore cancelling the proton charge very effectively

On magnetism you can also observe shadowing, only you have to cancel a dipole with another dipole. This is somewhat harder to observe at macroscopic levels as magnets are large and you would have to move far away, however we already have a good example of shadowing from different materials.

As you may know Some materials of the periodic table are ferromagnetic, which means that they are magnets and generate a magnetic dipole, however the atomic root for magnetism lies on the spin of electrons paring of the electrons according to Hund's rules. In a nutshell each electron carries a spin, which can be considered a small magnetic dipole. If you pair electrons with different spin, the net magnetic dipole is cancelled, but if you put several electrons with the same spin, the total magnetic dipole is increased.

As you may have guessed at this point ferromagnet have many electrons with the same spin, therefore each atom carries a large magnetic moment, which can add up producing large magnets we see every day. On the other hand many other materials do have all their spin paired (or almost completely paired) so the net magnetic dipole is neglegible.

Finally you may be able to "simulate" electricity using magnets, however this will rquire from you to somehow "simulate" magnetic monopoles, which can be achieved if you consider 2-D surface (such as a piece of paper) and you move magnets from the out the plane

I’m thinking about modeling electricity using magnets but I realized that a limit of that model is that you can’t have two “neutralized” magnets ... not be attractive to a third magnet...

You recognized that electric charges are separable but magnetic are not. Have a closer view on what electrons are and you’ll understand why’s this is so. Electrons have an electric charge and a magnetic dipole moment. Both this are intrinsic - under all circumstances existing - properties. The electron is a fundamental particle which one can annihilate with its anti-particle to pure energy for a moment. But the electrons electric charge and the magnetic dipole moment are not separable nor divisible.

BTW it is possible to design a magnetic generator and a magnetic device with “wiring” between them. The wires have to be of a big cross-section, say metal rods. On the generators side you install a cross with four arms with alternating north and south poles. Rotating this generator a cross of the same design on the over end of the metal rods will came in rotation.

Take four ordinary bar magnets and arrange them in a hollow square N-to-S so they stick together. Their attractiveness to nails is now sharply reduced. If you made them into a neat ring or toroid it would disappear altogether. So your initial assumption that magnets cannot be "neutralised" is wrong.

What happens with both the toroidal magnet and the hydrogen atom is that their respective fields get trapped locally and can no longer spread outwards indefinitely.

You may be thinking of the fact that the poles cannot be sawn apart into an N magnet and an S magnet, but that is a very different phenomenon.