The machine boots and runs fine from floppies.  Some things I've read indicate that it could be capacitors or the power supply.
If the system works reliably from floppy disk, but there's apparent SCSI issues, then power supply/capacitors are unlikely.  It's not as if the power supply magically knows if the system is booting from floppy disk or SCSI then somehow decides to deliver unreliable 5VDC or not.
If the hard drive LED is almost continuously on, and it's extremely slow, it may be that you're using using FastFileSystem, the volume has a validation error, and the system is scanning the entire volume as part of the validation/correction process.  If you boot with no startup-sequence, then just leave the system alone until the hard drive LED eventually goes off (might take an hour or more), does it eventually validate?
A SCSI bus termination problem can cause symptoms where SCSI transfers are unreliable, though interesting that the issues persist with a different SCSI controller and SCSI drive - are you using the same media on the different SCSI controller?  Are you able to test the same SCSI hardware and media in a different Amiga?
There's a small chance of a memory issue causing the system instability.  Booting from hard drive does use more memory (file system is in memory, plus disk buffers, etc), meaning it the system would be using other areas of memory that it it wouldn't be when booting from floppy.  An easy check is to use the AmigaTestKit bootable floppy, or 
https://aminet.net/package/util/moni/MemCheck12