I'm A Banana
Potassium-Rich
Though I've never thought that 5MWD was a significant problem, even in 3e, I'm fond of the way your conception of dailies as kind of "always on" abilities. For the goal of motivating PC's to continue on, though, this doesn't work very well at early levels, when they only have one daily ability, and it doesn't work THAT well even when they have more: "I have to use this limited ability!" isn't a reason to keep going. It prevents a nova, but it doesn't give them any reason to keep going: yeah, they're not out of juice totally, but why not rest and recover your abilities, even if you're not totally out of them?
In order to keep the PC's moving forward, you want to use positive and negative reinforcement: incentives that awesome things come to them if they forgo a rest, and consequences for taking a rest that they'd like to avoid. The AP serves as the former, and if you'd like to get rid of that, you'd have to dangle some other carrot out for them: what do they GET, if they keep pressing forward?
Since this is a meta-game problem, you can probably successfully address it in the meta-game: bigger rewards (more GP or XP) for continuing on, for instance.
In order to keep the PC's moving forward, you want to use positive and negative reinforcement: incentives that awesome things come to them if they forgo a rest, and consequences for taking a rest that they'd like to avoid. The AP serves as the former, and if you'd like to get rid of that, you'd have to dangle some other carrot out for them: what do they GET, if they keep pressing forward?
Since this is a meta-game problem, you can probably successfully address it in the meta-game: bigger rewards (more GP or XP) for continuing on, for instance.