For me, the biggest thing is that often higher level = more tanky.
Take Fiora for example, a non-tanky melee champion. She gains up to 100hp per level (well, 99 for 17 to 18). 100hp is no small number, that's 2/3 of a ruby crystal. There's also the innate armor/mr/ad growth, but even just looking at health it can make a big difference.
So you have let's say a tank support. Despite itemizing to be a tank and being innately tanky, if he's 4 levels behind the solo lanes he's not going to be tanky at all really. His only real tank benefits are CC and tanking objectives (Dragon and Baron, towers to a lesser extent).
For someone like Janna, Soraka, Sona, etc... I don't think they necessarily need the level growth. Their kit is all about helping their allies from safety (and if they get caught, they're supposed to be blown up). It's the supports that want to be in the action but can't because of a level disadvantage that is problematic.
And you can't go and just give Braum, Leona, and company extra base health because then they'd become obnoxious top laners or junglers. So it's really a tough spot.