This is getting to be a bit of a strange argument on both sides, IMO.
I definitely agree that 6-8 encounters in a day is ridiculously high. I expect "most" tables are somewhere in the 3 encounter range, give or take 2.
However, I think that if you have less encounters, they generally last more rounds. I can't imagine a Paladin not being able to use up their spell-slots to smite, even with the new restrictions. And if they do have a few slots, well, they'll just be able to use them for spells.
I may be missing something, but the only thing the "nerf" seems to do (AFAICT) is to stop the Pally from blowing multiple smites in the same round. They can still smite every round if they like, which is some pretty darn good damage output. I'm not sure any class is designed to blow a spell slot every single round.
If you only have 1 encounter in a day, and it only lasts 3 rounds, NOBODY is going to use up much of their resources. I don't see why the paladin should have to be able to blow all its resources in that timeframe.