You are correct. The word "amplitude" is not used carefully enough here. The hopping has dimensions of energy. In perturbation theory for the wavefunction, one gets back a real amplitude because there is an energy denominator as well. So, if one were doing perturbation theory for the wavefunction, it would read as
|state> = |0>
+ sum_n |n>
The numerator, has a matrix element with value (what i called "amplitude") t. But then the dimensions work out because of the denominator.
I think this abuse of nomenclature is common. Strictly it is only an amplitude once the energy denominator is included.