Is it that protons tend to join other H2O molecules rather than forming H402+?
Yes. Think about the availability of water (its molar concentration) as compared to the concentration of H+ in a typical acid solution.
And water doesn't contain only H3O+. From what I remember also larger ions like H5O2+, H7O3+ and so on were detected.
Sounds like it does. It's just unlikely to happen in most solutions where the number of water molecules far outweighs the number of hydronium ions.
I'll also add that under almost all conditions, the formation of H2O2+ is probably very thermodynamically unfavorable compared to hydronium formation. This means that even if the concentrations were equal, the formation would still be less likely to occur due to instability of the molecules.
Think about it: why would a charge accumulate on one molecule when it could spread itself out?
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