More Schwinn bar bike refurb confusion - fork headset standards?

Okay, in the ongoing saga of fixing up a neglected Target level schwinn hybrid from "dangerous BSO" to "just don't go over 10mph and it will be safe enough to ride", I'd like to replace the suspension fork that sat for months with water inside the lowers.

Miraculously, it uses a 1 1/8" threadless steerer, and i've installed multiple threadless forks on mountain bikes with sealed bearings and matching crown races, so I have some knowledge there, but i don't understand how fork compatibility works with these caged bearing type headsets.

Could I buy any low end rigid fork (that matches the wheel size, brake mounts, steerer type, etc) and just swap it in (after cutting the steerer and installing a star nut)? Or do I need to find a specific crown race for my bike's specific caged bearings?