Am I reading this right? Phoenix deathflare is spammable?

Unlike Deathflare, brand of purgatory has no note that using it ends Firebird trance. You need a buff tho from casting fire ruin 3.

So you can go ruin 3 deathflare, ruin 3 deathflare rinse and repeat until FB trance done.

Or am I missing something?

I guess it is technically upgraded outburt but looks more or less like fire deathflare to me.

Deals unaspected damage to target and all enemies nearby it with a potency of 400 for the first enemy, and 50% less for all remaining enemies.
Can only be executed while in Dreadwyrm Trance. Dreadwyrm Trance fades upon execution.

If you are referring to Phoenix rotation, yes, it's going to be Fountain of Fire -> Brand of Purgatory and repeat, all while weaving the 2 revelations.
Unless my math is terribly wrong you should be able to do 4 fountain of fire and 4 brand of purgatory before Phoenix leaves the battlefield. If that's the case it's going to be 3500 potency with 4 FoF, 4 BoP and the 2 revelations in just 20 seconds, without taking into account the DoTs, which will of course add up a little bit. It will be an important moment of your rotation, so pray for the boss to not go immune or some bs.

Brand of Purgatory is 50 less potency than a Deathflare, but yes, it can be used more than once.

Fountain of Fire (Ruin III) into Brand of Purgatory (Outburst) spam and of course there are the two Revelations.

Gallus, I refer to this

Deals fire damage to target and all enemies nearby it with a potency of 350 for the first enemy, and 50% less for all remaining enemies.
Can only be executed while under the effects of both Firebird Trance and Hellish Conduit.
※This action cannot be assigned to a hotbar.

Rather than replacing Deathflare, Brand of Purgatory is replacing Outburst. Otherwise yes from footage we'll alternate from Fountain of Fire to Brand of Purgatory.

Brand of Purgatory Lv. 72 Spell Instant 2.5s

Deals fire damage to target and all enemies nearby it with a potency of 350 for the first enemy, and 50% less for all remaining enemies.
Can only be executed while under the effects of both Firebird Trance and Hellish Conduit.
※This action cannot be assigned to a hotbar.

Fountain of Fire Lv. 72 Spell Instant 2.5s 400 MP

Deals fire damage with a potency of 250.
Additional Effect: Grants Hellish Conduit
Duration: 10s
Can only be executed while under the effect of Firebird Trance.
※This action cannot be assigned to a hotbar.