Honestly, it sounds from what you people are saying that I should be looking for a different game system. I appreciate the suggestions from Rhenny on how to add more encounters and they would be good suggestions as to how to trick the players into playing differently. But I don't want to do that. I don't want a game where I have to keep having non-dramatic combat after non-dramatic combat. I want a game where combat is an episodic climax, dramatic and challenging. Not where it is dramatic and challenging one game in six.
You can, of course, do that in 5e, there's just a few factors you have to take into account. One is whether you (or your players) care about class balance at all. If nobody does, then all you have to worry about is keeping encounters challenging and workable even at 1/day. That's still a challenge, since you'll have everyone throwing down their best tricks, quite possibly, every round of the fight, but it's possible, and you've gotten some good advice on how to swim against the 'fast combat' current built into 5e.
If you do want to impose some class balance in spite of consistently falling far short of the expected 6-8 encounter day, though, don't panic. The classes aren't balanced to begin with, you would have had to do some tweaking and spotlight-roving to achieve it, anyway in reaction to party composition and other player choices, anyway, so it's just going to be party of the usual task of DMing. Just with a different focus.
Party composition could even solve the problem for you: if you end up with a party where everyone has some potent resources to 'nova' in their daily uber-encounter, for instance (or if no one does). If you get the more typical mix of classes, though, you need to forget about resource-management pressure over a long 'day' as a check on the heavy-daily-recharge classes, and find other ways to limit them, and build up the other classes. Rather than focusing on resource pressures to do that, latch onto other aspects of the disfavored PCs to help them shine. If nothing else works, you can always give the overshadowed PC(s) a nifty magic item with an awesome daily function or few.
Honestly, this is mind-boggling to me that a role-playing game has such restrictions in the style of story you can run built in.
They're not restrictions, they're just degrees of freedom that have other implications than just the pacing of you story. You can think of them as 'imbalancing' classes or 'ruining' encounter-building guidelines, or you can think of them as tools that let you leverage and re-balance the PCs and the challenges posed by encounters to better fit your campaign.
I've played a lot of different RPG systems - started off with Shadowrun and White Wolf WoD games. Dipped into CoC.
Those are pretty narrow- style- and feel- focused games.
I've never found a game where I was forced to add low-meaning combats just so that the big meaningful combats worked mechanically. I'm finding all this very disheartening. My group expressed interest in playing a fantasy setting game so I bought D&D as the famous classic fantasy game. I'm honestly finding it good in some ways, but rules-wise it seems to be very shaky and odd. I'm about ready to look around for something else except I've spent a whole bunch of money on this product.
You've certainly come at D&D in an unusual way. Most of us stepped up to 5e with multiple editions under our belts and very clear expectations of what D&D was about. And that has always included managing resources over the course of an adventuring day in one sense or another. As was mentioned above, the closest D&D has ever come to being neutral to pacing issues (and it wasn't that close) was 4e, which put all classes on the same resource schedule and made the 'daily' portion of those resources relatively less significant, allowing the DM to have much 'shorter' days with no impact on class balance and less (though still quite significant) impact on encounter balance. 4e was not well received by the fanbase who came to D&D in the more usual way, with more traditional expectations, and is out of print and not on the OGL that allowed 3.5e to enjoy ongoing 3PP support (and, 3.5 if even further from what you seem to want than is 5e), so it's not the freshest horse to hitch your wagon to at this point. Indeed, as the old saw implies, we really should stop beating it at some point...
...and, as you say, you already sprang for the set of 5e books (the good news is that you don't have to keep buying new ones every month to stay current). So, you might as well make the best of the current, supported edition. IMHO, the best thing you can do when coming to 5e without much of a background in D&D is to get some experience playing with a long-time DM and get a feel for the system. You should have a much easier time of that, being experienced with other RPGs already, but there is an art to running D&D and it's best picked up by playing under a good DM, and, of course, by doing. Fortunately, it's very easy to find current-edition games, just look up AL events in your area....
Good luck!
