I see your point and it might very well work, but might just take a lot more work then fixing the support income problem, either could fix it and a special uility stat can be implemented after anyway and be worked on more slowly through new champs and reworks
My only issue with a utility stat is that it could still cause balance problems when supports are played outside of the support position
Lulu is a great example of that, and personally ive played champions like Braum, Taric and Alistar in top and jungle as well .
How would you see that work out, supports would gain their gold much quicker in such positions and with greater gold efficiency they could maybe snowball through utility or something
I had a game recently where an Ivern did this by building Redemption, Locket and Athenes, while the rest of us had 2 items at best and most of the carries only had one and a half item every time the Ivern ganked it was tower diving for days with the shields and heals and all the cc and slows
He had a lot of defensive stats and sustain for the gold spent and utility out the ass buffing everyone up and there was little our enemies could do to stop this
So that is something to consider, i am not at all against a utility stat, but i think it is an unlikely idea considering the need to do so much work on so many champions and then redoing and inventing new items, testing them and balancing them, that is preseason material ^^
However fixing support income is something which might be achievable before then after which it is much easier to fine tune the utility they have around stats they would already buy, and slow and steadily implement a new utility stat through reworks etc
However besides all this there is one other problem, build diversity for some of those supports could suffer.
Also Riot has been exploring ''utility'' stats, like % healing and shielding increase for example already exist
So this is something we will probably see more of in the future alongside new forms of utility
Which is something i would really like to see