To us 'get off my lawn' grognards, this seems a touch excessive.
I mean, I'm pretty most mere "get off my lawn" grogs are actually fine with it, as evidenced by this being just about the first time this has been suggested.
I think you're talking more about sort of "get away from my cave-entrance" levels of ultra-grog/grog-king.
As for your "nerf", well, for a lot of casters, it'll have low impact, because being real, most cantrips are cast less than 2x between short rests, and as you level up, would get less of an issue. I am assuming it wouldn't apply to Warlocks as they're their own thing, and it wouldn't make any sense for them.
The people it'll really screw over are casters who are intelligent and engaging with their game, or using cantrips for RP. They'll be stuffed by your idea, and suddenly people will be very reluctant to use cantrips and think about them less, and do less cool stuff with them. Do you want that? Is that a goal? It doesn't sound like a good idea.
One problem is that currently casters are not balanced for this - some casters have access to more cantrips. This would hit them less hard. So I'd suggest giving casters more cantrips (1-2 more at least) at L1 if you did this. That's more reflective of older editions anyway. Some cantrips are clearly not designed to be limited in usage either, so would need reworking.
The main result will be low level casters will all carry a light crossbow, which does more damage (at a slightly lower chance to hit) than these cantrips.
This assumes non-combat cantrips are meaningless, which they're not, and also that you're using all your spell slots in combat, which you're not.
No, he's right. Balance-wise this is extremely low impact, almost unmeasurably low. Role-playing-wise there'd be a bit of an impact, of a negative kind.