That actually is my preferred interpretation. "Distant Origin" wasn't Trek's first visit to this well; that was the TOS novel First Frontier by Diane Carey and paleontologist Dr. James I. Kirkland, which posited that a species of raptor dinosaur was transplanted Preserver-style and gradually evolved sentience on another planet. Since I count that book as part of my personal continuity (and have alluded to it in my professional Trek Lit), I assume that the Voth's hadrosaur ancestors were transplanted in the same way.
Still, as I said, it would be a mistake to assume it's impossible for an advanced civilization to have existed in Earth's ancient past. The popular conception is that we have a pretty full record of Earth's prehistory, but we only have isolated fragments; most of what we conclude about the past is extrapolated and deduced from those bits of evidence and from basic scientific principles, and we're always discovering new data that compel us to reconsider some of our assumptions.
Of course, they could both be true. A single species would be unlikely to survive 65 million years unchanged. That's long enough for a taxonomic line to produce dozens of "generations" of descendant species, the old ones dying out and being survived by the newer ones. It could be that a technological hadrosaur civilization evolved on Earth and traveled into space, left colonies, lost their civilization, evolved into new species, invented spaceflight all over again, migrated still farther across the galaxy, etc. until the Voth's ancestors finally settled in the DQ 20 million years ago. There could easily be more offshoots out there.