Well, the really interesting thing about the path system was that if you had a spell shared by two paths you could jump between those paths.
So let's imagine two paths. Fire magic and Light magic. They share Continual Flame as a spell. You could start with as a fire mage and then pick up Continual Flame to progress on the Light path, which would allow you to pick up Invisibility (assuming it is a Light spell). But you wouldn't be able to pick up Mage Armor, because it's not on either of those paths (you'd need to first learn a spell on one of your paths that was also shared by the Force magic path). Maybe not a perfect example, since Continual Flame is an illusion and arguably wouldn't constitute fire magic per se, but if that's the issue just substitute an imaginary spell called Everburning Flame that meets those criteria.
That's what really appealed to me about the path system. It built on itself in a logical progression. The principles of one spell allowed you to branch out to related spells.
There are some challenges to the path magic approach, particularly when it comes to distinctions between Arcane & Divine magic (as well as subtler class spell list distinctions) that are baked into D&D. For example, running with your Fire (Pyromancy) path example...
We see 3 cantrips that fit –
control flames (druid, sorcerer, wizard),
fire bolt (artificer, sorcerer, wizard),
produce flame (druid).
Clearly
control flames fits a multitude of class archetypes, but what about
fire bolt vs.
produce flame? Does the path offer both cantrips? In that case, the distinction about "druid-fire" being able to be used to illuminate like a torch becomes shared by other arcane casters.
What about the cleric's
flame strike which deals both fire and radiant damage? Would that be included in the path, thus being equally accessible to Arcane and Divine casters?
Similar identity question about the druid's
flame blade...
That raises the question would this approach include two separate Fire-themed paths, one for Arcane Fire and one for Divine Fire (speaking conceptually, not saying those are good names)? If so, does the Divine Fire get called "Radiance"? But then you have
light & produce flame competing for the same design space. And would
flame blade (which does exclusively fire damage) be included in the Radiance?