I see what you're saying, Mort, and yes, D&D has long had a bias that spells > skills/ability checks when it comes to solving problems. That, again, however, isn't empowering a player unduly in of itself- spells are (mostly) a limited resource. You use a spell to circumvent a problem, that's one less spell you can use that day.
The issue here is, as Tetrasodium points out, that the DM has the power to keep spellcasters in check by making them loath to use their spell slots willy nilly. They also have the power to rule on how spells function (and I'm not talking about rewriting them, though that's also in their wheelhouse if need be, but simply interpreting what WotC wrote- and didn't write in spell descriptions). And finally, at least in the Wizard's case, they have the power to keep the Wizard from gaining a large potential spell list by limiting things like enemy spellbooks or scrolls, and not giving the Wizard the time or money to scribe new spells into their book.
But a lot of DM's don't. It's not terribly fun to force extra combats, or insist on X combats per diem. I don't know how common this is in the wild, but most of the younger DM's I know are inspired by video games and want to make these big setpiece battles, where getting more than two fights into a session is a challenge- ironically, these DM's might have been better served by 4e's design, in which most of your resources are per encounter, not game day, but that's not how current D&D really works.
So the number of spells you can use hews more closely to the number of combat turns you have, and once these big fights are over, everyone is fuming on hit points and wants to take a long rest. This results in spellcasters having spell slots laying around at the end of sessions, and it doesn't take much of that before caster players realize they have power to burn, so why not cast spells at every opportunity?
5e's design does not, in my opinion, overly empower the player (or their character). The DM still has all the power to set the pace of their game. The issue is, some DM's do not do this, because it's not very much fun. If you come into the game via public play, you'll see that when you play nothing but adventures you can knock out in a few hours, there's no way to drain everyone's daily resources, and so the modules don't even try- see above for how that affects spellcaster play.
At the same time, of course, these adventures give limited short rests, which puts undue stress on the short rest recovery classes; I've seen people try to play a Warlock in AL and get to cast one whole spell that wasn't Eldritch Blast per session.
As an aside, D&D One wants to address this by making short rest resources into "x uses per day", but I'll wager WotC will do nothing to address game days with fewer encounters.
The hit points and hit dice healing players have access to now are subject to the same problems as spell slots. The DM has the power to make players be more frugal about healing, and take less risks, only if they want to enforce a higher amount of encounters per "game day". This is good if you're tackling a long adventure and the average encounter is "you enter a 30' x 30' room. There are four Orcs here, seated at a table, eating some sort of stew, no doubt prepared in the kitchens of room 2G. They spring up and attack!"
This is not so good if the average encounter is "you exit into a large courtyard. The lizard priest is atop a raised platform, about to plunge a dagger into the heart of the captured elven princess to enact his ungodly ritual. He is flanked by four Lizardman Acolytes. Between you and the priest are six burly Lizardman Elites, and the Lizardman Commander, who is astride a Riding Beast. Large stone braziers sit atop 5' high pillars in the corners of the room, and the floor is a multicolored mosaic of large tiles, each with a particular significance to the the lizard folk- you know from experience that some of these tiles are trapped, but so far, the only color you know to avoid is fuschia."
So it's not that there is some imbalance in "power" for the DM, it's simply that the game is asking them to run adventures in a very particular manner, that even WotC themselves obviously finds tedious and boring, and they have yet to address this with even optional rules.