Having run a bunch of the Quickstart adventures, and played in a game that ended at level 3, I think the game absolutely can run a generic AP completely fine. As I've said (and I didn't invent this idea), if you run Daggerheart that way, you end up with a better 5E.
Running it much more free form is running it more as intended. The problem is that you have to be a pretty good GM to run games that way. Running a low-prep game that's truly reactive to the PCs is hard. I've played a lot of PbtA, and that's the playstyle it wants the GM to use, as does Blades in the Dark. Not all GMs can run a game that way. And (this is my opinion, of course) running a fully free-form game based on the players, and doing so largely off the cuff, is a terrible experience if you're not good at it. I've played enough PbtA at Cons to know that's true.
If you have the improv chops, go for it. Otherwise, running something more structured (such as a game where you get a setting, characters attempting things, and you interact with them) is something I expect most experienced GMs could do.
Whatever your skill level as a GM, and whatever your players want out of a game, I think Daggerheart can give you great sessions. You just need to pair them with your own skills.