Why can we use an infinitesimal switch on factor $e^{epsilon t}$ in TDPT to describe scattering?

I know that there are many other ways of describing scattering in QM, but the way used in our lectures was to describe the, in reality static, scattering potential as instead having a ‘switch on’ time dependence (i.e. replace $Delta H$ with $e^{epsilon t} Delta H$, and then using the TDPT theory framework and at the very end taking the limit of an infintely slow switch on i.e. $(epsilon rightarrow 0)$ effectively constant scattering potential.

My question is why do we do this in the first place?

Firstly from looking back over the derivation of TDPT, it looks to me like it is equally valid even for a time-independent perturbuation $Delta H$ (since it just uses the time-dependent schrodinger equation, and re-expresses a state in terms of states of the unperturbed Hamiltonian, then constructs a perturbative series in $Delta H$, which all seems like it should work in general). Am I correct in this assumption that TDPT is perfectly valid in theory for a time-independent $Delta H$?

Therefore it looks to me like the only reason of introducing this switch-on factor is so that in the second order corrections (shown below) we avoid the possibility of singularities due to degenerate initial states (i.e. if $E_k = E_i$ which is guaranteed to happen if our initial states are momentum eigenstates since they can be the same energy but in different directions.

$frac{d}{dt} |c_n^{(2)}(t)|^2 approx frac{2 pi}{hbar} delta(E_n – E_i) |sum_limits k|frac{langle n|Delta H_0 |k rangle langle k| Delta H_0 | i rangle}{E_k -E_i -iepsilonhbar}|^2$

So, in my opinion, the switch on factor is just a ‘fudge factor’ to obtain this $-iepsilonhbar$ term in the denominator with no physical merit. Is this true or is there a better reason behind it?

I’ve also that in more complex work the $i epsilon hbar$ is some sort of prescription telling you how to integrate around poles, but nonetheless am I right in this $e^{epsilon t}$ switch on just being a fudge to get this at least in this formalism?