I had a problem like this with one of my A1000's. I tried swapping out the drive for two different A500 drives but the A500 drives suffered the same problem as you are describing (They would start to read the disc, the system would reboot, and ask for the kickstart disk again :/).
Eventually I managed to find another A1000 internal drive and now seems to be working again. Not sure why the A500 drives wouldn't work though as they worked fine in my Amiga 500. If you can try to get your hands on another A1000 drive (not sure about the A1010 but it might be worth a shot) then hopefully you can get it working again.
Sorry I can't be more help.
