The knights also tend to glow like Kaladin did when he absorbed the stormlight at the end.
Careful there, it says their plate glows, not them. I'm not saying they didn't use stormlight, because they probably did. I'm just saying we've never seen anything that specifically proves that. We still don't know for sure why they were able to use their powers in plate when Szeth cannot, so clearly there is SOME difference there.
And yes there is certainly a reason why Elhokar's plate has an inordinate amount of broken gems, but his plate wasn't that seriously damaged so I don't think Dalinar was pulling from those gems. It seems to me that he would pull from the gems inside his -own- plate first, since they are closer to him. -Something- broke Elhokar's gems, I just don't think it was Dalinar.
We also don't know for sure that the system runs on stormlight. It could be more intricate than that. It's possible that the Spren require stormlight for some reason in order to provide the abilities to a person. But it's entirely possible that this wasn't always the case....off the top of my head I would say perhaps when the almighty was around he was providing the power directly to Spren to act as vessels for who ever they were bonded to, but when the almighty died (and his shard possibly splintered) the spren no longer had direct access to his power. So now they require "batteries" in the form of gems infused with stormlight in order to provide those benefits.
Sorry long winded, but my point is that there are other things that could be going on here, that's just one example of what it might be.