RAW, only the caster needs a clear path to the target
Based on your quote on spell targeting, RAW, it would appear that you must be the one to have the clear path to the target, since you're the one doing the targeting; the Wildfire Spirit is just where the spell originates from once cast, and apparently does not require a clear path.
This is probably because, when such rules were written, there was no way to cast a spell from somewhere else (except via a familiar, but only with Touch spells, so that already took care of itself), since to have a clear path to target the target would also result in a clear path for the spell you just cast to reach the target.
Odd scenarios arising from this RAW reading
However, this leads to odd scenarios whereby your Wildfire Spirit could be blocked from your target (say, it's in a room, whilst you and your target are in a corridor just outside the room), and yet, you could cast something such that the spell originates from the Wildfire Spirit going through the obstruction at the target that you have a clear path to but your Wildfire Spirit does not (i.e. the target has total cover from the Wildfire Spirit).
For example, say you cast thorn whip at the target in the corridor whilst your Wildfire Spirit was still in this other room, which would then appear to leap out of the wall from the target's perspective, presumably then pulling the target into the wall.
The rules on cover (PHB, p. 196) says (bold italics emphasis mine):
A target can benefit from cover only when an attack or other effect originates on the opposite side of the cover.
A target with total cover can't be targeted directly by an attack or a spell, although some spells can reach such a target by including it in an area of effect.
So that first part of the quote implies that the "thorn whip through a wall" thing wouldn't happen, since the effect of thorn whip originates from the other side of the wall, but rather the thorn whip would just slam into the wall inside the room where the Wildfire Spirit is instead.
However, the second part of the quote that describes what total cover actually means talks about "targeting" again, simply stating that you can't be "targeted" by a spell, but if the druid and the target in the corridor have a clear path, then it can be targeted. That implies that, RAW, the odd thorn whip thing can happen after all, even though this certainly isn't RAI (Rules As Intended) or RAS (Rules As Sensible).
How to rule this sensibly as DM
It seems as though a DM would need to adjudicate this based on common sense. In other words, it's likely that for this to work, a DM could require both you and the Wildfire Spirit to have a clear path to the target; you to target it, and the Wildfire Spirit so that the spell's effect could plausibly reach the target.
However, this wouldn't matter as much if the spell itself didn't require a direct path (at least from a narrative perspective), such as frostbite, since I, at least, don't imagine that spell as having particle effects flying out of your hand and onto the target in the same way that I imagine ray of frost would look (although in that case, why bother making the spell originate from your Wildfire Spirit in the first place?).
Ultimately, it's up to the DM to make sense of this on a case-by-case basis.