I'll be honest, I found it really tricky to follow the OP's explanations, which I don't think is anyone's fault as the smite logic is complicated. I think I might not agree with their description of how it is working, but it's equally likely that I've just misunderstood.
I'm not sure it's any clearer, but the following gives my understanding of what I think is happening based on level 4 (ie one attack) paladin using Searing Smite with the game set to ask for Divine Smite on crit and non-crit attacks.
If I'm right about how it's working, then I don't see anything obviously problematic. It would be great if someone could point out what is wrong, or not consistent with 5e RAW. Or indeed help fill in the details that I didn't manage to capture.
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Paladin uses Searing Smite (SS) full attack (requires them to have an available action, bonus action and spell slot)
Then
- If attack misses
---- No weapon damage is applied
---- Action is consumed, bonus action and spell slot are NOT consumed
---- Paladin can do something else with bonus action if they wish
---- THE END
- Else if attack hits
---- Paladin's action, bonus action and spell slot are all consumed
---- Weapon damage is applied
---- SS fire damage is calculated and applied
------- If the enemy dies THE END
------- Else SS condition is applied to the enemy* and to the paladin
------- If there are any spell slots left, user gets asked whether they want to use Divine Smite
----------- If user says yes
--------------- Divine Smite damage is calculated and applied, and spell slot consumed
--------------- If the enemy dies, SS condition removed from paladin THE END
---- Paladin's turn ends
---- Then the following REPEATS repeats for 10 rounds including the current one **
------- Each time the paladin gets a potentially concentration-ending effect then they need to make a save
----------- If they fail, SS condition is removed from paladin and the enemy THE END
------- On turn of an enemy affected by SS condition
----------- SS damage is calculated and applied (doesn't consume any paladin resources)
----------- Enemy gets opportunity to save against SS condition ***
-------------- If they succeed, SS condition is removed from the enemy and paladin THE END ****
------- If an enemy affected by SS is killed, SS condition removed from paladin THE END
------- On the paladin's next turn
----------- They can use their action, bonus action and remaining spell slots however they wish
----------- If they use any new concentration-using spells/effects, SS condition removed from them and enemy THE END
----------- Else go on REPEATING until the 10 rounds are up
* I didn't write down whether the enemy gets an opportunity to save on first application of SS condition, I think they might.
** I'm not sure how the duration is being calculated exactly as I've never been able to get anywhere close to exhausting it. I guess SS might remain active until the start of the paladin's 10th turn after their initial use of Searing Smite, or until the end of their 9th subsequent turn, or until the end of the enemy's turn where they get SS fire damage for the 10th time (including the first hit).
***I didn't write down whether the save was before or after the damage was applied on the enemy's turn.
**** I'm not 100% sure about whether the SS condition was removed from the paladin straight away as I found it really hard to maintain concentration long enough to check, and a couple of occasions it looked like they still had SS but with duration reduced to 1 turn. I guess it's possible that the paladin could retain the condition and have the opportunity to reapply next turn, but I didn't see this and I think all that was happening was that the paladin was losing the condition at the end of the enemy's turn if they made a save.
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EDIT: Btw, though I don’t find anything obviously bad about the mechanics of SS it does seem conceptually a bit weird. It’s an attack that can set your enemy on fire and potentially keep burning them as long as the paladin keeps focusing. It would make more sense to me if it were an attack that set your weapon alight and gave it a chance to do fire damage for 10 turns or unless concentration was lost, though I suppose then it would make less sense for it to be tied to whether or not the first attack hits.