The entire point of wards not having their own slot is so you have to give up vision for your full build.
As such, trinkets give you some limited help, but the entire point is to not let full build people light the map up like a christmas tree.
and why shouldn't people be able to do that?
Because it removes any single advantage a behind team has, vision wise? If your full build, the max wards you can have out is 3 (assuming not a support); 1 pink, 2 greens. If the pink is destroyed, you can have 2 greens, unless you trade trinkets.
This means full build teams either have to give up some power for vision or have less vision overall. Due to this, teams that are not full build (behind teams) could have space in their inventories to ward, giving them an edge in vision to try and catch others out, surprise, or flank the ahead team.
Making everyone have a slot for wards would mean a full build team could ward the entire map, with the behind team then being behind on damage, vision, or both, instead of just one.
Correction though, 3 wards per person with 1 vision ward per person.
Besides its not like wards totally don't cost money. Most behind teams are going to leave vision to their supports, and continue trying to rush full build anyways. And the team ahead? Well, the jungler and support will likely grab sweeping lens tinkets, the jungler will probably smite the chickens and rip all those wards are now cleared out.
Anyways this would motivate people to actually buy wards, even late game, or midgame(if you're fed to that point where you have full build). I'm lucky if I see anyone other than supports warding(sometimes not even them, and the other person is usually the jungler).