YSK: Cats don't like their whiskers touching their food bowl. This is why they beg even if food is still left at the bottom.
Whisker fatigue causes stress and overstimulates their senses. Using a flat bowl or plate will relieve this issue and allow your cat to eat all their food without stress.
You can also search for "whisker fatigue" bowls specifically made for cats.
While whisker fatigue is in debate, it's true some cats are weird about touching the bowl. I personally feed the cats on the same plates I eat from and they get water from a fountain. But I have known many cats who were fine with a bowl too. I think the main thing is the material of the receptacle. Some plastics can hold bacteria that makes their lil faces break out. :(
I also use the same plates we use, but they would still ask for me to throw out the food and serve a new batch everytime (I don't actually throw out the food but 'recycle' it).
This is my experience, too. We've had a lot of cats (many rescues), and some really do have problems with the deeper bowls, while most others don't. We got some shallow bowls years ago, and those work well. We tried just using small plates, but a lot of cats will push the food over the edge and make a mess, so some sort of rim is important for us.