Yes, the origin of the non-zero rest masses becomes open again if the Higgs mechanism is not responsible. But it will almost certainly be something similar, if not specifically the Higgs then some equiavlent or condensate (possibly from some new phsyics, which is why it would be exciting); there are too many constraints from the physics we do know for it to be a completely open question.
Good point. Would it be fair to say that mass is the result of a particle and its accompanying field even if it isn't the Higg's particle/field.
Also, finding the Higgs is no barrier to investigating new physics. On the contrary, the Stanard Model with Higgs is still the most bizarre hodge-podge and is a blatent sign that it is merely an effective theory of some deeper physics.
I can completely agree with that. That was the distinct impression that Weinberg (speaking of the electroweak force) made on me when I read "Dreams of a Final Theory".