There's that, but I mainly referred to the fact that there are only a handful of utility items to begin with. You have the Crucible, Twin Shadows, the three support items, Randuin's Omen, Aegis/Locket, Frozen Heart, the Censer, the Herald, the Sightstone and that's pretty much it. Most supports will only be able to effectively utilize only a handful of these items (Tanks: Omen/Frozen Heart, Face of the Mountain. Mages: Frost Queen's Calm, Twin Shadows, Crucible), and the Censer is most effective on supports with crowd shield/heals (Soraka, Sona, Thresh).
My point was that support champions mostly have to rely on whatever their kit can do rather than depend on their items to provide that utility for them. Any champion can support in theory (when I started supporting, I was using Galio, and earlier this year I liked to use Volibear), but items don't provide simple things like stuns, and the Crucible has too long a cooldown, even with the appropriate masteries, for the heal to be in any way practical; it's basically a third summoner spell. By default, you're put at a considerable disadvantage as a non-support champion in that role because many non-supports simply lack that type of utility in their kit.
Case in point, a few games ago, I went support Thresh with Vayne against support Lee Sin with Graves. Even though they had early bursting power, it didn't take long before I started stacking on enormous amounts of armor (a personal objective in a game as a tank is to have over 100 armor before level 11), and their ability to jump in and bully us out of the lane suddenly fell flat. On top of that, while I could throw down lots of stuns and slows and provide lots of shielding, Lee Sin simply couldn't keep up with the demands in an all out brawl, and he and Graves were quickly overwhelmed.
while true that any non-support can go down there part of the problem with the thresh lee matchup was thresh increased armor and cc vs ad damage and only minor slow and ult knockback. he was doomed from the start against that combo.
also in my OPINION that was a kill lane not a "support" lane. many champions who have actual cc can go down to "support" and become realistic champions later, such as annie (who was never classified as a support type, unlike lux or morgana). there are quite a few of those champions, annie(stun passive), fiddlestyx(fear and silence), arhi (charm), anivia(stun, slow, and wall), cho' gath( knockup, silence, tanky), mundo[kinda](heavy poke, tanky), elise(stun, heavy poke), heimerdinger(stun, heavy poke, zone control), jarvarn(kockup, tanky, wall ult), jayce( heavy poke, ms buff, knockback, slow), jinx( heavy slow, snare, AA poke, finisher ult), kennen(manaless, good poke, stuns, aoe stun), kog' maw(% hp dmg poke, slow, vision ult), leblanc used to before Q change, lissandra(slow, snare, CC ult), Malphite(cc ult, slow, tanky), maokai(heavy poke, knockback, snare, aoe dmg redux), nautilus(hook, slow, knockup), panth(tanky, stun), ramus(tanky, taunt, thornmail, speed), ryze(tanky, snare), shen(taunt, tanky, shield), shaco(fear, trap vision, escape), singed(tanky, slow, flip, chase prevent), skarner(tanky, capture, slow), syndra (stun, slow), talon could before silence remove, teemo( trap vision, blind, poke), tristana( slow, poke, knockback), twisted(stun, map vision, slow, poke), urgot(slow, poke, dmg redux, translocate), varus(slow, %hp poke, aoe stun), vel' koz( knockup, slow, heavy poke), vi(armor redux, tanky, knockback, suppression), victor(snare, heavy poke), voli (tanky, flip), xerath( slow, stun, finisher ult, heavy poke), zac(tanky, knockup, slow), ziggs(knock back, slow, heavy poke, finisher ult).
as you can see it is a large list. all of these champions are capable of "supporting" and some work as a straight kill lane as well.