I would tend to assume that food waste should mostly be compost, as long as plastic wrappers and other non-biodegradable waste is kept out. Which, depending on how it is implemented, moving away from trays would seem to be a step backwards, to me.
Hardly at all.
The real reason to remove trays from all-you-can-eat dining is to save money on ingredients - force customers to choose between a drink and a dessert and you'll have to pay for far fewer desserts.
First, you clear the land for farming, which includes machinery use, so transportation of petroleum using petroleum, then burning it, repairs and manufacturing of machines, etc.
Then you do the farming, which involves a bunch of petroleum burning for machinery and transporting the things needed to grow it.
Then you harvest it, again, same thing, plus storage.
Then you transport it to where the grown things are processed.
You do the processing, package it up, and transport it to the dining hall.
Where it's thrown into the trash.
Emitting carbon which recently came from the atmosphere back into the atmosphere is not a problem. Emitting carbon which has been safely stored in rocks for tens (or hundreds) of millions of years is the problem. That's the carbon that's caused the increase in CO2 from 280ppm 200 years ago, to 420ppm today, which is the driver of climate change. Nothing to do with food waste.