The Schrodinger equation describes particles, and right back at the beginnings of quantum mechanics it was an obvious question whether a field could be quantised in the same way. The first steps I know of in this direction were Born, Heisenberg and Jordan's paper Zur Quantenmechanik II in 1926, so it really does go back almost to the beginnings of quantum mechanics.
It only took a few more years for physicists to realise that a field theory could describe both fields and particles, that it could do so in a relativistic way and that particle creation and annihilation emerged naturally from a field theory. So quantum field theory was a no brainer. It was an elegant idea that described a wide range of physical phenomena.
The early problems with field theory weren't with the basic ideas. The non-interacting scalar field theory is elegant and exactly soluble. The problems were that the equations describing interacting fields were very complicated and the perturbative approaches used at the time didn't work. They produced the infinities that you allude to in your question. Renormalisation hasn't change the basic ideas. It just means we now know how to do the calculations correctly.
Is there any indisputable evidence the field is second quantized?
and the answer is that quantum field theory is tested every day in colliders across the world and has so far proved effective at describing the behaviour of fundamental particles. So that would be a yes then.
One last comment: the term second quantisation is an unfortunate one. It's original meaning was a second method of quantisation i.e. an alternative approach to first quantisation, and not that anything is being quantised twice. Few physicists I know would use the term second quantisation because of the potential for confusion. However you will still find it being used disappointingly frequently.