Searing Smite analysis

One of my gamers plays a multiclassed level 3 paladin / level 13 fighter. In this edition of D&D, paladins gained a new spell that applies minor (1d6) instantaneous fire damage and adds ongoing minor damage until the target makes a saving throw, which ends the spell. The gamer questioned whether multiple melee attacks would add more ongoing damage. After my analysis, I find the current ruleset is unclear. The following is my interpretation:

The proper interpretation is not to have the ongoing effects stack.

The target is burst 1d6 extra fire upon each attack. This is an instantaneous effect and it is applied for each attack. (It could be viewed that the initial 1d6 extra stacks.) Each melee attack also applies the "OMG I am on fire as a secondary effect but this effect does not have a name" secondary condition. These secondary conditions do not stack. Once the target has the condition, they have it, and they don't get another from more attacks.

(reference: While this is not specifically addressed by the stacking rules which are limited in scope to benefits and penalties, the designers have restricted stacking rules more in this version than in the past. Even METEOR SWARM explicitly does not stack with itself in 5ED, although it explicitly has stacked with itself since Expert and 1ED.)

So the secondary effect is SV VS CON (DC 8+5+5=18) or take 1d6 fire damage, success ends the spell, repeat each round.

If Cazzian wishes to overload the secondary 1d6 damage he can hit multiple targets. Of course they would each get a saving throw, and any one of them making their save would end it from that point on.

Interestingly, the SV VS CON DC 18 should probably be 8+1+5=14 due to multiclassing into a non-casting class, but whatever it's another flaw.