I've never seen a single cleric player who wanted to do that, though. Most of them prefer the option to cast a variety of spells in a variety of situations. They'll use bless or the like at times, but it's not remotely the default assumption.
Familiarity/contempt, I guess. Clerics have had a lot of options for a very long time. Until late 2e or 3.0, though, they mostly 'had to' use their spells to heal. When wands could handle healing, the gloves came off, and CoDzilla rampaged for years. Then they became more reasonable again.
Fighters couldn't do support much at all, when the Warlord finally enabled that, going all-in was a natural reaction, I suppose. Maybe we just haven't had enough time to get sick of it yet? The longer it takes to get a Warlord into 5e, the longer it'll be before we do.
Maybe that's part of why some people are having trouble getting their mind around the warlord? (Again, just theorizing.)
It's a nicer theory than most...
Clerics and bards can be played as primarily support characters, but they don't have to be. So what else does the warlord do besides just support?
There's a few things he shouldn't do: obviate the fighter by being just as good as him whenever he wants, for instance.
Apart from that consideration, some things the warlord has done or could do that might not be strictly support:
- Discourage enemies from moving.
- Draw an unsuspecting enemy out of position.
- Force an enemy back with a fierce attack.
- Frighten a group of lesser enemies into hesitating (not moving or not attacking on their next turn).
- Foil an ambush.
- Confuse an enemy, lowering his initiative, or limiting his actions on the next turn.
- Coordinate volley fire or a formation attack (possibly of NPCs), resolving it quickly like an AE, instead of each subject (or the DM) rolling each attack individually, and make it more effective than the individual attacks would have been (again, especially for NPCs). (It's the old, train the villagers to defend themselves thing.)
- Use a reaction to bring an ally to his rescue when attacked.
- Trick an enemy into a trap or AE of an ally (ok, the second part is kinda support).
- Drive a wedge between two enemies, so they attack eachother.
- fake a command from a Boss, causing a lesser enemy to take an action of his choice
- Disarm an enemy
- Find a flaw in/damage an enemy's armor or other defenses
- Find a 'counter' to an enemy's attack, rendering it less effective for the rest of the battle.
- Tricking an enemy into a 'false sense of security' so it's initiative is delayed, or it doesn't take critical action immediately.
- Trick an enemy into revealing a weakness, or deduce the weakness if INT-based rather than CHA-based.
- Make a devastating 'Surprise Attack' against an unsuspecting enemy who thought he was just a caddy.
- 322* other things you might lift from 4e...
- heck, 756, we'll throw in fighter exploits to really avoid support (it's not like the 5e fighter's use'n 'em)...
- like, goad/trick every enemy w/in 10 or 15' to move adjacent, then whack 'em all with one great cleaving blow (or a series of quick Inigo Montoya thrusts).
- And, at the risk of causing some heads to explode: Pull a 'big reveal' where some 'plot point' or reversal of fortune changes things - an enemy is actual a sleeper agent that infiltrated the bad guys years ago, the ground in front of the charging force was undermined by deep gnome allies no one knew about, re-enforcement come riding over the hill. The whole melodrama thing. Yes, it's an /actual/ 'plot coupon,' and yes, it'd be double-dog/triple-pterodactyl, hidden in the highest-walled, most inescapable of Option Ghettos, quick-avert-your-eyes, whole-table unanimous approval required, you will neverevereverevereverEVER have to put up with it nor even hear of it actually happening if you don't like plot coupons, OPTIONAL.
But it's fun at tables that can handle it.
Some of them are even D&D tables.
(That's part of the meaningful choice I was talking about earlier.)
Seriously, though, the big thing missing in the handful of 5e sub-classes that don't wield supernatural power, so far /is/ such choice. You can multi-attack or SA or (maybe) rage and do a lot of damage, or you can do something else you'll have to wrangle with the DM over to 'improvise' that is unlikely to be effective as just dropping the guy in front of you. Having a lot of choices of how to do some 'support' for your party would still be a step up.
*Now, I know that someone's going to pop up and say "but all those kewl powerz only work because 4e was so grid-dependent!." The grid is just 5' squares. Last session, my Warlord 'slid' a monster 6 squares so it'd be in the Burst 5 my wizard ally was casting. Just because it's 30' instead of six squares, and 'trick into moving' instead of 'slide' doesn't mean it can't happen in 5e. In 5e, it wouldn't even /need/ to be 30' it could just be "into the area of the ally's attack" and the DM could rule how far made sense to him, if he'd even been keeping track of positions so precisely while running TotM as to worry about it. (In 13A, it'd just be "add one to the number of enemies the triggering ally can target." Now that's TotM.)