Do lower control arm reinforcement plates actually work?

Depends on the type of driving you do.
If you drive like an old lady probably not but to like pushing the cornering ability especially with sticky tires then it becomes more of a must. You don't want a squirmy car in the corners at high speeds.

But it's one of those things why not do it cause it doesn't hurt to.

Plus all the little things add up. Is it gonna give you the biggest reward for effort no.
Tires and sway bars fall into that category.