There are several confusions here. First, you seem to be mixing up particles (described by infinite-dimensional unitary irreps of the Poincare group) with fields (described by finite-dimensional, generally nonunitary irreps of the Lorentz group). I'll just focus on the particles, but for more on this point see this question and the linked questions there.
Now, there's a simpler objection that is equivalent to yours. The Schrodinger equation says that for a time-independent Hamiltonian, each energy eigenstate just rotates its phase as $e^{- i E t / \hbar}$. Since these phases just rotate independently and uniformly, how can anything nontrivial ever happen?
The reason is that the energy eigenstates for any nontrivial system are extremely complex. For example, an unstable particle can decay, but that's because the particle itself is not in an energy eigenstate. The true energy eigenstates of the system are exceedingly complicated superpositions of the particle and its decay products, which nobody can compute. Thus, conversely, starting with an initial condition of just the particle present actually corresponds to taking an exceedingly complicated superposition of energy eigenstates, with their phases aligned just right. As the phases start to rotate, their relation to each other changes in time, causing something nontrivial to happen.
In principle it's similar to, e.g. normal modes in classical mechanics. Many linear nondissipative system in classical mechanics can be written as a bunch of independently oscillating normal modes, but that can give rise to complex time evolutions.
In the case of the Poincare group, the situation is just the same, since the heart of your complaint is just about the factorization of the time evolution operator. But you might be wondering, don't textbooks seem to imply that the time evolution is more complex than this? Yes, and the reason is that in general, they break the Hilbert space into Poincare irreps under the free time evolution, thereby defining "in" and "out" states. (Here, "free" is defined as any Hamiltonian that is simple enough so that you can actually perform this procedure.) By definition, these states don't interact with each other, but when they get close, the full time evolution takes over. The effect of this time evolution on the free irreps is, of course, described by the $S$-matrix.
In summary, the "true" Poincare irreps indeed have "trivial" time evolution, but we see nontrivial time evolution in practice because these irreps are related to the free Poincare irreps in an exceedingly complicated way. Textbooks always construct the free Poincare irreps, because it's not feasible to say anything about the true ones.
