You could cheat a little.
Select just a face and extrude-tapered. A "Perfect point" will fail and disappear, but you can zoom in and stop just before it is a perfect point.

In real life there is no such thing as a perfect point.

Then you can separate all the pieces and show points and just grab and stretch the bottom points down for as tall of an extrusion as you need.

Here's another way to do it as well.

Maybe advantageous to have a straight cutting edge inside the cookie cutter ?

I like this one vic

And then the machine mechanism would have a little “pimp walk” on the way up… seperation!