Yes, it does seem logical but there are several factors that you haven't taken into account. The first is that "H₃O⁺" is a "formalism". Not long ago, chemists would just write "H⁺" as "H⁺" and leave it at that. However, that implies a "naked" proton floating around in solution and no one actually believes that is what actually occurs. So, if it isn't a single proton then it must be something else. But what? The simplest assumption is that the proton combines with H₂O to give H₃O⁺. That does the trick. But is that what actually happens in solution? Good question. There is a significant amount of evidence to suggest that the proton is actually coordinated to four water molecules - that it is actually H₉O₄⁺ (pictured here) that is the active transport species. That is certainly consistent with the remaining elements of the group (i.e. [Li(H₂O)₄]⁺ or [Na(H₂O)₆]⁺) and would simply solvate the proton like any other cation.

But there are also arguments against this idea as the proton moves through solution much faster than might be expected for such a large species. Then again, there are arguments that it is not the same proton that moves through solution. Rather, a proton is shuttled from cluster to cluster, in an atomic equivalent of the old "bucket brigade". What the actual chemical species that is present in solution is, well, still a question of debate although I would suggest that most chemists feel comfortable with the H₉O₄⁺ species.

So, why not H₄O²⁺? Well, there are electrostatic arguments. Putting positive charges together is energetically expensive. There are space arguments. The proton once it is "coordinated" to the water molecule is no longer "naked". It has electrons which means that it is quite large - certainly as large as the other two protons which help to define the tetrahedral structure of water molecules. But likely the most important consideration is that water is 55.5 Molar. Even at an acid strength of 1 M, there are 55.5 water molecules to each proton. That is a lot of competition and it is unlikely that you are going to find protons doubling up! After all, it is energy expensive and there is no need with all of the available partners.

Hope this helps.