Do you still want to install 7.1? Version 7.5 and newer got rid of the enablers, so you'll have no problem with those.
These are actually the disks downloadable from Apple's server and since my Centris doesn't have a CD drive, I have to resort to floppy disk images. System 7.5 images aren't available on the Apple server and I've been highly unsuccessful in trying to get System 7.5.3 files on the Centris.
It's perfectly possible for a disk to be bootable like this, with the contents of the system folder just poured into the root level of the disk. Technically the root directory is the system folder then. I'm not sure if Apple would have ever made a disk like that but I guess it's possible.
It sounds to me like the disk has been altered from its original state. I think a possible answer would be to find an image of the first disk for system 7.1 and write over your disk with it.
As I said earlier in this post, these are the disks straight from Apple's server. Either way, I re-imaged System 7.1 and copied over the System Enabler and it seems to be working now, so thanks for all your help.