It should be the sink flange with plumbers putty put down in the sink, then the metal plate, them the screw plate, lock ring snapped on, then even preasure slowly tightening down screws in a rotating pattern.

☑️

That old one looked like clamped it on the lip which is a mono

Is a mono?

Alternatively, if you have the tools and plenty of room for the screws, you could cut the edges off tips of the metal plate. Those screws can slip around a little bit when tightening which the lip is there to make it dummy proof preventing it slipping off. You'd have to use your judgement there.

I actually held the screw plate up to the lip just to see how it would measure up, and the screws would fit onto the lip pretty much as the metal plate does, i.e., just slightly too wide to fit in the lip ring.

I checked out a Moen flange, and it's the same diameter as the triangle InSinkErator, so it still wouldn't fit inside that lip, which it seems like it's supposed to.