Take a look at 13th Age. It's a d20 game from Rob Heinsoo (lead designer of 4e) and Jonathan Tweet (a lead designer of 3ed). It came out before 5e, but both share a lot of the sensibilities in terms of streamlining, some similar and some done in other ways, plus it adds in some of the more current industry sensibilities that D&D really couldn't as part of it's brand. It has a good amount of 4e concepts, though since 4e wasn't under the OGL the names have changed (Recoveries instead of Healing Surges, Staggered instead of Bloodied, etc.)
It keeps the idea that powers are A/E/D (and some fun variations on those), while not making the classes move in lockstep in order to attain inter-class balance. Instead it has a full heal-up (equivalent of a long rest) happen every 4 encounters, not tied to something at least partially under player control. (Players can take one sooner at cost.) So with that in place it can have fun - for example there are no set cantrips, but some spells are At-Will, some are per Encounter, and some are "Daily" (full heal-up), so you pick as you want. Every spell can be upcast, so if you want an at-will 7th level magic missile, go for it.
Anyway, if you're looking for something with modern 5e-like sensibilities yet incorperating a lot of the good bones of 4e, take a look at it. The kickstarter for the 2nd edition is just about to launch, but you can see all the existing rules for free in the SRD.