Is forcing forced on us?

[Timothy Y. Chow]

I have recently been thinking (again) about giving a motivated exposition 
of forcing.  In particular I asked a question yesterday on MathOverflow:

https://mathoverflow.net/q/369710

In the course of that discussion, I was reminded of one of the dreams in 
Shelah's essay "Logical Dreams": Show that forcing is the unique method in 
some non-trivial sense.  (Non-triviality is of course needed since forcing 
isn't literally the unique method, but I think it's fairly clear what 
Shelah is trying to say.)

To a mathematician who is accustomed to ordinary mathematical structures 
but has no prior familiarity with axiomatic set theory, the forcing 
machinery seems like an inordinately complicated way to extend a structure 
(a countable transitive model M) to a slightly larger structure (M[G]). 
I am trying to get a better sense for why the axioms of ZF seemingly force 
forcing on us.

It occurred to me that if we throw out enough axioms of ZF then eventually 
we will reach a point where can can build new models more easily.  So my 
question is this: is there some weakening of ZF that is

1. not so weak as to be uninteresting, but
2. weak enough that some new (i.e., not forcing) method emerges to 
construct nontrivial models?

Tim