Why star PCBs?

Newbie question:
why are high power LEDs mounted on star PCBs with 6 contacts, rather than one rectangular PCB with just one + and one - contacts?

This allows easier placement.
Lots of pills and shelfs have holes for wires, the star shape lets the wires get to the pads
Some pills/shelfs have holes for screws, again, using 1 or 2 cutouts of the star still allows attaching the wires.

The hex footprint also allows the stars to fit together and daisy chain them in series. Parallel wiring was frowned upon for some time because variations in Vf could cause one led to experience thermal runaway. It’s not much of a concern anymore with DTP copper and more consistent Vf’s. Still, mixing LEDs of different types requires separate current control for each type.

I still don’t get how rectangular shape would prevent from doing all things you described.
Maybe the “original” shape was circle, to better fit into flashlights? And making holes around a circle results in… an hexagon?

Dunno, but hex was the more common variety with wire and screw slots being interchangeable. Even Blackburn mtb lights had stars several years ago. Both Cree and Souel were mounted on them. Maybe the metal die punches had an easier time with 60 degree corners than with 90, who knows?

I think the hexagonal shape is the most versatile.
Fits in a circle, press fits easy because the points will give way and it will still centre properly, can be mounted with 2 or 3 screws, meanwhile leaving openings for the wires.

Less waste material when punching them out?

Also, typically just the 20mm ones come in that hexagon shape. Many smaller lights use smaller stars…. typically 16mm or smaller.

These smaller stars usually come in a circle shape with two slots and 2 contacts.