There's 3 big problems with utility scaling:
- Utility automatically scales in how it's used.
A 1 second stun when you're the only person to capitalize on said stun is only 1 second of extra time for you. However - a 1 second stun when you have 4 allies nearby is a collective 5 seconds your team has on the stunned champion, which is usually more than enough time to kill a rather fed but squishy champion.
As LoL tends to balance champions on their power over the course of an entire game, you could very well make several utilities that are good in team-fight scenarios almost completely useless early game.
Which would make several supports 100% useless for the baby-sitting a weaker champ in lane aspect of why they're picked.
- Utility has been used as a way to allow champions with low gold/levels to maintain a prescience in the fields of justice since pre-season 1. Without scaling, a 2 second stun always being a 2 second stun means it's 1 seconds early, 1 seconds mid game, 1 seconds late game. You don't have to worry about being phased out or useless even if you can't tank/damage things in a meaningful way.
With scaling, it's also worth noting that a
build for a support (if they even get enough gold) isn't exactly uncommon at the competitive level. I've seen that run on Nami a couple of times already, and an important thing to note about it - 0 AP.
Beyond AP, such a build mostly focuses on supporting and utility over stacking any single stat like HP, AP, or AP. The only stat support builds tend to amass a large quantity of is %CDR.
With scaling - you also kill the other key aspect of supports over a strong early: Being good with minimal income. If supports need income to actually support, then they're useless in lanning and useless late without items.
- If the scaling is solid and worth pursuing - you send that champion in a farm lane. So they have the gold to get the items that boost that scaling:

A lot of supports don't have much scaling because a reliance on scaling is anathema to every reason to use a champion as support in the first place.
Which also does a decent job at explaining why a lot of mages (with no reliance on scaling for their CC/utility) started booting out a lot of traditional supports when Rito decided Sona, Lulu, Janna, and Soraka should have AP needs tied to their utility. Then Lulu and Soraka took over mid lane because hey, better scaling than mages >.<