So I created a 5e version of the Warblade that uses a resource that is unlimited, but not usable at-will. It can be found here:
http://community.wizards.com/forum/dd-next-general-discussion/threads/4058286
D&D is filled with examples of limited resources for non-magical characters. Even in AD&D there are fighter kits and other martial classes with daily abilities. In 3e there were feats, rogue tricks, barbarian rages, and many prestige classes that granted daily abilities.
I personally dislike "daily" abilities in general so they never really worked for me, but limited resources like that have existed in D&D for a long time.
I like the way 13th Age handles limited use abilities for fighters and rogues with random trigger abilities and momentum. The rogue's momentum abilities, if ported over, would make for a very interesting 5e rogue.
I also used a variant in 4e that made "powers" a little more believable. Instead of daily and encounter powers, you received 3 red poker chips and 3 black poker chips. When you use an "encounter" power, you have to spend a red poker chip. When you use a "daily" power, you have to spend a black poker chip. You can use the same power multiple times per encounter or day. A short rest recovers all your red poker chips and an extended rest recovers all your black poker chips.
Whenever you reach a milestone, you can choose to regain 3 surges, a black poker chip, or an action point. Milestones were given out by the DM, not every 2 encounters.