I agree with all of the design goals cited by the OP.
I'm not saying you don't, but I feel the need to mention that a common and effective rhetorical device is to begin by agreeing with your opponent in the debate, then refuting everything you just agreed with, point-by-point.
5e needs to include balanced classes,
It also needs to offer distinctive classes that don't feel shoehorned into a single bland template,
I also agree that class balance is important. /Every/ version of D&D has had classes that were quite distinctive, but, the only version of D&D that has ever delivered class balance is the one that put all classes in a consistent mechanical framework.
The idea that doing so makes the "feel shoehorned into a single bland template," is, because of the choice of the word 'feel,' of course, a matter of opinion or reaction, and not something that can be used as a design guideline, since it's going to be different for everyone. Without the word "feel," of course, the claim that a common mechanical structure is a 'single bland template,' is simply false. If a common mechanical structure rendered everything using it the same, all spells in AD&D, for instance, would be 'the same.'
a clear and coherent design and purpose,
a design and purpose that doesn't elevate one portion of the RPG community over another
That all depends on what you think 'elevating,' means. If you have a clear and coherent /design/, that delivers a balanced game, then, by definition, it doesn't 'elevate' any one play style, since any given character or choice is reasonably balanced. A wealth of meaningful, viable choices ("balance") supports the maximum breadth of styles but /not/ elevating any one of them.
D&D has, for a long time, catered to some quite specific styles, giving mechanical rewards for 'system mastery' or, before that, for an 'exploration' style of play that emphasized paranoia and planning.
have teamwork baked into the system,
have teamwork baked into the system without requiring or favoring the use of miniatures,
I'm not sure where having teamwork baked (inextricably) into the system is absolutely desirable (there are some very small groups out there, and requiring 3 or 4 or 5 distinct character types to be able to face the full range of challenges would hamper them). Rather, teamwork should be something that the structure of the game facilitates, rather than requires. For instance, making the potential contribution of each class to a team quite clear is helpful in facilitating teamwork. Conversely, having each class able to contribute meaningfully most of the time helps a very small party get by without having a full slate of specialists.
That shouldn't have anything to do with the use of miniatures, and no extant version of D&D requires the use of minis, let alone makes teamwork impossible without them.
offer options and variety in customization rather than providing a fixed set of rarely-changed daily/encounter powers
Variety in play is certainly desirable, it's one aspect of balance - offering meaningful, viable choices in play as well as at chargen/level-up. Certainly, customization is desirable, a class should have choice when built, when it levels up, and in actual play. Furthermore, to maintain class balance, there should be a rough parity in the number and importance of those choices among all classes.
A common structure for gaining abilities does that, particularly if it discourages, minimizes or even bans taking the same limited-use ability multiple times (as that increases variety in play, /and/ reduces the impact of the inevitable marginally-superior choices).
and provide ease of play without assuming players can't perform simple mathematics or remember basic details about commonly used spells.
I don't know why you single out 'spells,' specifically, as any class might have some commonly-used abilities that become familiar in play. But, in any case, even though there is a strong (and perhaps deserved) stereotype of our hobby as appealing to 'nerds' who have a stereotype of being 'good at math,' there are avid gamers who suck at math or for whom a lot of math (or more likely arithmetic) slows or detracts from play. Complexity and simplicity are both, in a sense, desirable - complexity for what it /delivers/ more than in itself, while simplicity is desirable in and of itself. It makes sense to make the game as simple as possible, while retaining balance (lots of meaningful, viable choices) and interest.
and feature distinctive and interesting monsters.
And fairly easy to design and run encounters, which means a dependable system for pegging the challenge posed by given monsters.
Creating a system for those who want simple character sheets, enjoy mini-based tactical combat, and want to "not open a rulebook in a year" (as the OP stated) is great -- that's a community that was clearly underserved in editions past. Creating a system that also serves the needs of those with different preferences is the challenge with which the 5e team is currently confronted, though.
Indeed. The mandate to try to appeal to all fans of the game is a very high bar that WotC as set itself. Particularly, as segments of the fan base have defined themselves by edition, and gotten caught up in the specifics of their favored edition, both good and bad. Insisting on the bad parts of a favorite ed and demanding the scrapping of the good parts of a dis-favored one, will get in the way of a functional 5e that could be both balanced, and appeal to everyone. Teasing out the genuinely-good, vs the merely familiar, and the genuinely-bad vs the merely novel, would give us a better picture of what 5e needs to be.
Getting there would require some introspection, something that's tough, and not exactly what anyone wants to engage in over a mere hobby. So, my guess would be that 5e will fail, because, while it /could/ lift the best aspects of each edition and create a superior game, it probably (a) /won't/ as it pendulum-swings away from the best aspects of the latest editions (3e & 4e, that is) and (b) be rejected for excluding the /worst/ of any edition, because flaws are as precious as features to the committed edition-specific fan.
