(How) Could we discover/analyze NP problems in the absence of the Turing model of computation? From a purely abstract math/computational reasoning point of view, (how) could one even discover or reason about problems like 3-SAT, Subset Sum, Traveling Salesman etc.,? Would we be even able to reason about them in any meaningful way with just the functional point of view? Would it even be possible?

I've been mulling on this question purely from a point of self inquiry as part of learning the lambda calculus model of computation. I understand that it's "non-intuitive" and that's why Godel favored the Turing model. However, I just wish to know what are the known theoretical limitations of this functional style of computation and how much of a hindrance would it be for analyzing the NP class of problems?
You may wish to look at cost semantics for functional languages. These are various computational complexity measures for functional languages that do not pass through any kind of Turing machine, RAM machine, etc. A good place to start looking is this Lambda the Ultimate post, which has some good further references.

Section 7.4 of Bob Harper's Practical Foundations for Programming Languages explains the costs semantics.

The paper On the relative usefulness of fireballs by Accattoli and Coen shows that $\lambda$-calculus has at most linear blowup with respect to RAM machine model.

In summary, on this other planet things would be pretty much the same with regards to NP, but there would be fewer buffer overflows, and there wouldn't be as much garbage lying around.