He grates for a tomato sauce, which makes sense, but you wouldn't want that as your default onion cut.

Is there really a default onion cut? Desired sizes vary vildly depending on what you are making.

No, but it would have taken a lot more words to convey the same idea, so I'm happy just giving the gist. Most of the time, you don't want to grate your onions.

This is the MPW way:

https://www.youtube.com/watch?v=UBj9H6z6Uxw

"Perfection is lots of little things done well."

It avoids the issue except for the remaining bit which you can't grate without risking your fingers.

For garlic, I prefer crushing them for many recipes. This creates much rougher outlines that blend better into the food and crisp nicely when fried.

At that point why not just grind them in a food processor?

That's going to end up with a slush of onion fibre + onion juices. Very much different from even small bits of cut onion. Some recipes call for blended onions though.

Some food processors with the right setting give you chopped food not blended

A good food processor will cut the material, not blend it.

“You can grate the onion, or not, its your choice really”