USB Not Detecting

First thing I would recommend to check if it's a hardware USB subsystem problem or a disk access issue.

1. Check > About this Mac > System report... > Hardware > USB. If your drive is nowhere to be found then it's a hardware problem. Check your cables, try different ports, exclude any USB hubs etc.
2. If your USB drive is listed there launch Disk utility and see that your volume is listed on the left. Try mounting it.
3. If it's listed and not mounted chances are that you have some old third-party filesystem driver (like Paragon NTFS) which is not compatible with 64-bit only Catalina. Try uninstalling and upgrading the filesystem driver.

I had a somewhat similar experience with the aforementioned Paragon NTFS which has a free version bundled with Seagate drives. In my case other disks were mounted but in read-only mode.

When you close the lid of the computer your machine's USB goes into low power mode.

When you let your machine fall asleep due to Energy Saver Settings it goes into low power mode. This mode does not go back to normal until your machine is restarted.

To avoid this problem go to Apple menu -> System Preferences -> Energy Saver, and turn off put hard disks to sleep when possible, and turn on prevent computer from sleeping automatically.

Use Apple menu -> System Preferences -> Screen Saver, or the dimming display keyboard options to dim your display when you want to save energy, to avoid having to plug back in USB items all the time.

Shut down your machine before transporting it and removing devices.

Use an AC/DC adapter on a USB hub to improve connectivity. Some hubs only are powered by the computer, and that is often insufficient.

If all of this does not work bring the machine in to service to have its battery replaced.

It may also be cheaper just to get a refurbished Mac of that age if you depend on software that only runs on it.