Dragons melted the castle at Harrenhall (seen in season 2 or 3), so I don't think it's too much to assume it could melt the wall. Yes, the physics don't exactly make sense, and the half-cocked plan was ill-contrived, but it does move the story forward. I can suspend my disbelief a little more for good TV.<I think the damage the dragon did to the wall was a little over the top>
I actually did not understand what the dragon was supposedly doing to the wall. I guess the wall is supposedly made of ice and some rocks and not masonry or reinforcement? 700 feet tall. That is one massive amount of ice. I do not think anything we had seen from the dragons previously indicated their fire generated the kind of therms it would take to literally melt something like that. I mean previously dragon's fire burned up some soldiers and wagons. It did not turn the earth below into glass. If the idea was that the wall was not literally melted but that it fell down, what would be left, a 300 foot mound of slippery ice? Not something easily negotiated, I would not think. Melting a hole through it makes a lot more sense to me, too.
The entire conversion of a dragon seems contrived anyway. Going out to capture a wight to show Cersei never made any sense in the first place, and losing a dragon to the other side because of it just seems too pat. As it turns out, apparently, everyone would have been perfectly safe behind the wall for an indefinite period of time but for losing that dragon. There was no urgency to prepare to fight at all.