Yes, it's true , it happens a lot with HP printer drivers...
My guess is that you accidentally removed the drivers from Programs and Features...or they got messed up somehow...
What I did recently to solve that was like this:
1. I downloaded the HP drivers executable from HP.
2. I manually extracted the executable to a folder using
WinRar (The HP executable is actually a compressed file, so you can extract it like it was a .zip, .rar file...etc)
3. I connected my printer, I went to Device Manager, right clicked the missing/yellow questionmark item concerning my printer and selected "Update Driver".
Now on the update driver wizard, I selected Install driver Manually , choose location, and I entered the path where I extracted the HP executable.
Found my drivers, installed the printer,then tried to start the HP installation, which failed again, but the printer worked!
See if this helps.