I had the same experience with my Laserjet P1102w. Downloaded the printer driver named hp_LJP1100_P1560_P1600_Full_Solution-v20180815-50157037_1.exe
which was dated June 2021. Double-clicked on the .exe file and followed its instructions, My devices and printers found it, and I was immediately able to print out a page.
It's pretty shameful that Microsoft's and HP's support teams are either too uninformed or just unwilling to tell customers how do do this. It was really no different from installing the printer when it was new.