UESP sites Reman as being the one who began the lighting of the Dragonfires ritual. It appears to be from this line in http://www.uesp.net/wiki/Morrowind:Varieties_of_Faith....

The Dragonfires existed before Reman's time as both http://www.uesp.net/wiki/Lore:Trials_of_St._Alessia and http://www.uesp.net/wiki/Lore:The_Amulet_of_Kings refer to the Dragonfires being made when Alessia made the pact.

Hestra, an Empress from the First Era, performed the ritual as her soul is shown to be in the amulet in http://www.uesp.net/wiki/Lore:Where_Were_You_..._Dragon_Broke.

So was the lighting of the Dragonfires not a thing that every Emperor did before Reman?

The capacity for Empire Reman demonstrated defies any previous usage of the Dragonfires. The covenant started with Alessia, and was renewed by Reman, and again by Septim. In all honesty, the custom most assuredly began with the Ayleids.

If true, them thar fires weren't actually fire.

Remember all those are written by mortals, could be the two books claiming it was around during the Alessian empire are wrong or Varieties of faith has misinformation. Misinformation is a thing.

The Amulet of Kings seems to just paraphrase The Trials of St. Alessia's version of events, so I would just consider that to be the same source written by a later scribe. I have no idea if The Trials of St. Alessia is to be trusted over Varieties of Faith (probably, but who knows?), but I don't think that Hestra's presence in the Amulet of Kings makes any argument one way or the other. Just because the Amulet was present in the First Era (and no one has claimed otherwise, Reman-fires or not) does not mean that the Dragonfires were also present with them. I have no real opinion on the matter because I've never been as interested in the Dragonfires as I have been in the Amulet, so I always ignore them when I look up stuff relating to Alessia, and have never really considered them in the context of Reman (because there's so much other, more interesting stuff that Reman is better-known for).

It could be that the way Reman-ritual is just for initiation purposes whereas Alessian-ritual was more religious.

The Red-bauble is a simple tool Akatosh uses to uphold his currently preferred flavour of stasis. The fires are, quite apparently, irrelevant as heat sources. They're the visceral expression of Preservation within and without the Empire's Domain. I say 'Empire' as a term for any of those who rule from White-Gold. Furtherblather, the capacity the Bauble as a 'soul gem' is tertiary, at best.