@Mike
Yes, they were worse.
Maybe it is the entire Cyberstorm Card and not the 060. That seems more likely to me, but an 060 is easier to try.
Did you try OS3.1 alone? Simplest thing to do would be a Workbench 3 boot floppy with the 060 lib(s) on. Work with a copy obviously, and of course you'll likely have to delete some files to make a bit of room first. Then run software from your hard drive (let's assume your software is virus free). Bare in mind your scsi.device version though and your partition setup etc...I can't be held responsible for any data loss if you directly or indirectly write to your hard drive! I'll assume you already have all of that in mind though as I know you're a clever chap.

For novices, perhaps that 911 boot disk would be of help here? I know literally nothing about it except for its existance, but it's likely got all this stuff on.
Unless you cold boot from another device (like floppy) then you really can't rule out Virus infection nor just corruption of some of your system files. But yes, that 060 would need the 060 lib(s) to fully test. In my gallery there's a VirusZ entry which lists all the files needed to fully check your system. It's worth it as I had a link virus infection called 'Ebola' a few years ago on one of my Amigas... VirusZ fixed it.
Out of curiosity, what is wrong with your onboard 030? Does it exhibit the same behaviour you are now seeing on your 060?