Ok… so after bit more of theory crafting and minor testing with the Fire Pets concept, I have found out that I don’t like it… 
Everything I could come up with, Booby Trinkets simply seem to do better. And If I wanted single dmg type, there is the trusty old Kelp Rings for Lightning Pets.
Dracarris could have worked well with the Fire Pet Rings, but its proc is on Enemy Death which only triggers if the player gets the kill, which is pretty much impossible on a pet build.
Maybe Witching Hour could work, but one could just go full on Chaos instead with it. Also given the CD on the skill + only having 50% Phys > Fire Conversion from both the rings combined, I am doubtful if Fire Witching Hour would outperform an Elemental pet build.
Another potential candidate is Skeleton Pet builds, but I have zero interest in putting myself through such torture.
If Doggos were good, then that would have also worked. But Doggo being more of a derpy, that doesn’t work either.
Also, if you go fire Birb, you only get 1 birb. That might be ok with the birb Conduit since it gives the projectiles the ability to pierce through, but that doesn’t help against Single targets.
You could go with Beastcallers instead, but there is the above mentioned thing about Trinket or Kelp rings. There is also the option to forget about all that and just go Bleed Pet build.
Do remember that Trinkets also convert Acid to Elemental and Every Occultist Pet build uses Blood of Dreeg while very few would have enough points for Hellfire.
Currently, I’d say that the only thing Fire Pet Rings have going for them is being Target Farmable. But given the low Drop Rate, I am not sure how well that works in practice.
That said, if someone manage to make them work well enough to actually compete with Elemental/Lightning builds, do let me know for I would be very much interested.