Unable to recover old documents from a Time Machine to my MacBook Pro

I want to recover old documents from a Time Machine I used on another Mac

It appears that you are confusing Time Machine.....which is software used to back up your Mac to another device.....with a Time Capsule......which is hardware.....an Apple AirPort router with a built in hard drive that is used to store backups of your Mac.

When I connect the two devices via a USB-C to USB A cable, I don't see the Time Machine on my MacBook

That's correct. The USB port on a Time Capsule is used only to attach an additional USB hard drive for additional storage capacity. You cannot connect the Time Capsule to a Mac using this type of USB connection....as you have discovered.

The Time Capsule is a "network" drive that must connect to the network using a wired Ethernet cable connection. In most cases, you would need to connect the Time Capsule to your Mac's Ethernet port.

If your Mac does not have an Ethernet port, then you will need a USB to Ethernet adapter.....or....you could connect the Time Capsule to one of the LAN <--> Ethernet ports on your router.

The Time Machine will appear under the Locations heading on the left panel of any Finder window. Click on the name of the Time Capsule and an icon representing the Time Capsule hard drive will appear in the window to the right.

The default name of the Time Capsule hard drive is "Data".

Edit to post above.....

The Time Machine will appear under the Locations heading on the left panel of any Finder window......should read

The Time Capsule will appear under the Locations heading on the left panel of any Finder window.