I think the Divine's writ was sufficient to get things moving initially. Whilst the nobility might have baulked a bit over it, the ordinary people would acknowledge its authority, which is how they initially get recruits, plus everyone being so upset about the Divine being blown up and those with any capability wanting to do something about fixing the hole in the sky. Then this gets backed up by this mysterious figure dropping out of the Fade, with the soldiers saying there was a woman in the Fade behind them, which everyone assumes is Andraste. This is reinforced when your first attempt at closing the Breach at least stops it growing any more. At this point the Inquisition has the popular vote and who cares about the Chantry or nobility because they are doing damn all about anything. It is noticeable how often people see you close a rift or succeed in solving a problem and the response of people is that obviously the Maker is with you, even if you strenuously deny it. As far as they are concerned, you are the Chosen of the Maker/Andraste and this shows the Divine's writ had to be genuine and the Maker/Andraste approves.
Then after you succeed in closing the Breach but Corypheus appears on the scene, then it is hardly surprising that the Divine's writ is no longer considered of importance in garnering support in view of the immediate threat. By this time, Josephine's diplomacy is bearing fruit and so people seem happy to let you get on with it. You still, of course, have the popular vote.
I was a little disappointed that "the will of the people" didn't have more bearing on events in Trespasser but then you have to consider that Andraste was meant to have had thousands of followers and yet that didn't stop her getting burned, so the Herald coming out of it still alive was a bonus. I don't think the former Divine's writ really had any bearing on the issue, other than being brought up to say "business as usual" if you decided to put it under her control or as a symbolic "that's what I think of your authority" as you drop it on the ground if you disband. Then again, I always opt for the aggressive option with either since the words pretty much sum up how I felt about the whole thing. Clearly from the scene that follows at the very end, you aren't giving up on the Solas issue whatever they think or regardless of official authority on the matter.