And a smite spell would be discharged by the first hit of course.
Not the way divine smite works in the 2024 rules, no. You hit the target, then immediately use a bonus action to do the smite damage. A DM might rule that since the casting of divine smite (explicitly) and the second attack (implicitly) must both immediately follow the original hit, you can’t use both.So the extra smite damage doesn't carry over?
Smite now triggers as a bonus action, for 1 hit.So the extra smite damage doesn't carry over?
Divine Favor is a better example for paladins.A spell like Spirit Shroud, which applies to all your attacks, would apply to both attacks though.