I think this is getting a little away from my main point: I don't think CS is currently flexible enough to extend it into a general physical combat mechanic. If it stays a Fighter only mechanic, perhaps it works (though I still have questions about balance), but if you want it to become the physical equivalent of spellcasting, it falls short.
Heh, you can tell a mechanic is popular when a thread explicitly AGAINST it instantly becomes a discussion about how to extend it to other classes.
As for the concern expressed in your OP that even with CS a fighter has very limited options, I think some perspective is important here.
A starting fighter has 3 uses for his CS die, giving him (at most) 3 class-specific options each round.
By comparison, a level 1 cleric has 4 spells (including a domain spell and Turn Undead), 1 Channel Divinity, and 1-2 orisons for a total of 6-7 options per round max - but since almost all his orisons are typically useless in combat, that's probably 6 options max. If nobody in the party is hurt yet, we're down to 5 (4 if the cleric also prepared Cure Light Wounds or if he's not fighting undead, 3 if both of those). And since those spells/channelings are daily, by the end of the day he could have a grand choice between attacking with a weapon and casting Radiant Lance.
The rogue? Two semi-class-specific options: he can sneak attack or he can hide. (The latter isn't strictly class-specific, but he's really good at it thanks to Skill Mastery.)
The arcane spellcasters are a bit better off: 6 options for the wizard (3 orisons and 3 spells), 5-6 for the sorcerer (two cantrips, two spells, an origin power, and a beefed-up melee attack if you want to count that), up to 4 for the warlock (although some builds would be as low as 1). But of course the majority of wizard/sorcerer options dry up by the end of the day.
Now, by level 5 the fighter has five CS maneuvers, which he can use in various combinations - up to 25 possibilities.
POSSIBLE POWER COMBINATIONS AT LEVEL 5
Fighter: any 2 out of 5 (all non-daily)
Rogue: 2 (2 non-daily)
Cleric: 14 (3 non-daily); but can prepare different spells each day
Wizard: 12 (3 non-daily); but can prepare different spells each day
Sorcerer: 11 (2 non-daily)
Warlock: 9 (9 non-daily)
So depending how you want to count fighter powers, fighters could have the most options at level 5 or second-to-least at the start of the day, but they almost certainly have the most options by the end of the day (except possibly the warlock).
Personally, I'd say all the classes feel a bit anemic at first level. A warlock who picks social and exploration invocations is pretty much an Eldritch Blast spambot; a cleric can toss out a couple flashy spells before he's down to basic attacks each round; a rogue begins his long career of spending combat trying to hide from (or gang up on) enemies and then stab them in the back.
However, I think this is largely by design. You don't want to overwhelm new players, and you want to encourage them to think freeform rather than just look at the powers listed on a character sheet each round. They also level up quickly enough early on that they shouldn't get too bored with that handful of basic abilities. The real question is, how much do those characters grow over the next few levels?
I think the rogue is a real potential problem, because while there is a good amount of strategy to positioning and so on, they could definitely use some cool combat tricks to make them more lively at middle levels.
The warlock is another issue, because the current power list contains a grand total of two attack spells (plus a sweet mass charm); it's very possible to have a 5th-level warlock in the current playtest whose total combat options consist of eldritch blasting or running away (in a very cool fashion).
The fighter? Honestly, I'm not so worried. Sure, there are adjustments to make, but the underlying setup seems fine.
Extending CS to other classes? I'm against it for a variety of reasons. Basically I'm worried that a 5e rogue using his half-ass CS dice would feel like a 3e ranger using his half-ass animal companion: they're so far behind the curve you wish the designers had just cut them out and made the core of the class more powerful.