Some games I have had progress on backing up, others won't back up due to a well known error, taken directly from Brad Taylor's FDS doc:

Software disk copy protection
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Special thanks to Chris Covell for bringing this to my attention.

Apparently, some FDS disks implement a very simple copy protection scheme, which the game relies on in order for the game to refuse to work on the copied disk. Normally, the number of files that exist on an FDS disk is stored in the second block recorded on it. However, some games maintain "invisible" files, which are basically files that exist beyond what the file count number in the file count block indicates. This poses somewhat of a problem for copy software like FDSLOADR, since these tools rely on the file
count block, and don't assume that there is any valid data past the last file found on the disk. This means that when these types of disks are copied, the invisible files will be lost, and when the game loads the files that do exist, the game's going to give the user heat about there being a
file missing or somthing, gumming up the works. However in practice, when an FDS disk is programmed, the unused end of the disk is usually completely zeroed out, and this makes detecting the end of the disk simple: just wait
to find a GAP period of extreme length. Except in rare cases, this model for detecting the true end of an FDS disk should generally provide the best results for copying the complete contents for all types of FDS disks.
To be honest, to back up a disk, you don't need this translated doc, as defeating the copy protection is only needed when writing to disks. To back them up, you'll need a tool like Brad Taylor's FDSLoadr.

Tomy of Tototek will soon be releasing a product similar to Brad's FDSLoadr that should address this problem. However, this bug described above does not prevent the technique described in the translated doc from copying from disk to disk or from using Brad's software and cable to write a .fds image to a disk. Also, did you know you can use Brad's cable to load an FDS image right onto the RAM Adapter like a flash cart? Technically, brad's design calls for two cables: one for loading games directly to the ram adapter, and one that hooks into the disk unit itself--this cable is for dumping disks and rewriting them.

The cable in the translated doc is for disk-disk copying, as opposed to Brad's cables.

Now, about your question on the belt. First, you must be positive the belt is identical to the original...no room for micky mousing on that (same diameter, width). Second, there are some adjustments you can make to the drive to get it recalibrated once the belt has been replaced. I intend to make an FAQ eventually, but right now, here's the closest thing that I remember seeing online:

http://members.lycos.co.uk/tomyweb/repair.htm

Now, I have never had any luck adjusting the motor of the FDS as shown on that site (I think I've made it worse doing this), but you may have better luck trying that other screw first....tiny adjustments only, as I understand it. I honestly have never successfully fixed an FDS drive, but armed with the knowledge in that link (and finally having a copy of Copy Master), I'd probably be able to get very good at it.

Chances are, there's nothing unrepairable in the drive, and it just needs some adjustments, so don't junk it.

-Rob