Apple TV 1st Gen

Are you sure you model is really 1st generation? Model A1218? The big silver-white slab of a box? Identify your Apple TV model - Apple Support.

The 1st generation (2007) has been obsolete since 2017. (1) It cannot connect to the iTunes Store anymore. (2) It can’t sync content through iTunes 12.6.3 or later. It can only work with non-DRM’ed content. No third party services. There is no support for it anymore. This model had it’s own software version path: the last supported is Apple TV Software 3.0.2 from 2010.

Apple TV with USB ports do not load anything by themselves from USB sticks. Restoring requires connecting to a computer. Then the computer (through iTunes) restores system software on the Apple TV box. Current software may not support this for obsolete models.

Apple TV has a USB 2.0 port. Your Mac has USB 2.0 ports. That is how you connect them. No need for Firewire cabling.

You would need to use a Mac with iTunes version before v12.6.3, as included with OS X Mavericks 10.9 or earlier, for you best chance. Lion would be just as good for this.