This is the recommended way to backup a website here:
Downloading an entire website over FTP can take some time, but you can generally just queue up the files and let FileZilla do it’s thing. You can also speed it up by increasing the number of concurrent connections from the default of 2 to the maximum of 10, which really makes the transfer five times as fast.
There is no other backup method I can recommend. If you’re having trouble using FTP, then let’s figure out how we can solve that. FileZilla has very good troubleshooting options, so there being “no error” can only mean you don’t know where to look.