Your insight about forces carried by massless vs. massive particles and their respective ranges is absolutely correct. Congratulations!

(Now, I'm wondering if this particular idea implies that since gravity's range is infinite, that implies that if gravity is transmitted by force-particles rather than space-curvature (assuming that that's a distinction with meaning), then the virtual gravity force-carrying particles have to be able to have arbitrarily small energies, and thus no significant rest mass...)

It is generally agreed that the still-to-be-constructed theory of quantum gravity will have gravitons, particles carrying the gravitational force analogous to photons for the EM field, and yes, gravitons should be massless as you argue. This is not however in conflict with the description of gravity as space-time geometry. Though the full details will have to wait till we understand quantum gravity completely, provisionally we can make unambiguous sense of gravitons at the pertrubative level: Think of a gravitational wave as a small ripple in spacetime, then one can quantize this perturbation and gravitons are to the wave as photons are to classical EM waves.