There is never going to be an answer to this debate as eye test is individual and the stats don't support inferior or superior chances within the box, at least at this point.
But these charts do help in at least proximity to the net for shot attempts.
Dallas Game
Eye Test - Flames iffy start, finish first strong, get down two in the second and pour it on.
Five on Five

Five on five I think makes your point. The Flames have the edge in scoring chances but they don't really concentrate their shots in the dangerous areas. Five on five chances were 8-4 Flames.
All Situations

Here the Flames dominate, showing the Bishop effect in the game. The Flames added nine more chances without being five on five (shorthanded chances and powerplay), and they were very concentrated in close and dangerous. This is Bishop stealing a game.
Colordao Game
Eye Test - Flames aren't the better team in the first but come out with a lead. Avalanche take it away in the second, Flames down push hard in the third.
Five on Five

Heat map shows a pretty even game, but the Flames broke down and gave up too many huge chances. Heat map shows a lot of the Flames shots came from well out, but they did get some in close chances. Flames gave up only four five on five scoring chances all game, all in the second period.
All Situations

In all situations the Flames had a 14-6 edge in scoring chances, and the map reflects many were in tight. However score effects are the difference as the Flames had a 8-1 edge in scoring chances after the Avalanche fourth goal. They didn't deserve to win this game, but Rittich wasn't great.
Ranger Game
Eye Test - Flames all over the Rangers in the first, but lose their mojo giving up an early 2nd period goal. Give up way too many chances in trying to tie it up. Better third, but lost in the second.
Five on Five

Graph fits the eye test as the Rangers have the better five on five chances. Flames go for it, and give up too many odd man rushes and breakaways, heat map reflects that. Chances are 18-14 Rangers at even strength.
All Situations

Flames are 7-0 in scoring chances in other situations, which catches the heat map up overall making the game look a lot more even than the shot clock would have you believe.
All in all these three games have the eye test match the stats. And by looking at heat maps I think the "Flames get weak chances in the box while giving up strong ones" suggestion is at least somewhat challenged.