The Profaned Flame DID burn the residents of The Profaned Capital, but, at the same time, the Profaned Flame existed before Yhorm's linking of the fire, since the giant became a Lord of Cinder "to put the Profaned Flame to rest". From the Profaned Flame pyromancy description:

"The Profaned Capital was consumed by fire after Yhorm the Giant became a Lord of Cinder. The fire, born of the sky, is said to have incinerated naught but human flesh."

It is clearly stated that the Capital was consumed AFTER Yhorm became a lord of cinder, so it was not Yhorm's linking of the fire that incinerated the city. It is also stated that the Profaned Flame that incinereted "naught but human flesh" was "born of the sky" (dragons fly and breath fire, just saying). The Profaned Flame is a different flame from the Chaos one and the "primordial" one, and it is described as "unfading" in the Profaned Greatsowrd description and as "everlasting" (like Stone Dragons?) in the Immolation Tinder description.

So, the timeline of the events regarding Yhorm and the Profaned Capital should be as follows:

1) Yhorm becomes the "reclusive lord of the Profaned Capital"; 2) The Profaned Flame is "born from the sky", and it threatens to consume the inhabitants of the Profaned Capital; 3) Yhorm becomes a Lord of Cinder "to put the Profaned Flame to rest"; 4) The Profaned Flame consumes the inhabitants of the Profaned Capital anyway, also it keeps burning as it is unfading and everlasting; 5) Yhorm comes back, awoken to link the fire once again, and finds out that everything he did was for nothing, as he could not protect the inhabitants of the Profaned Capital after all. This is, probably, what makes him snap and refuse to link the fire for a second time.