logitech harmony 665

As with any of these cases, check first for software updates from the vendor—Logitech, here—and apply those, and then ask the vendor support for assistance should there still be issues.

Running a web search for the vendor and product name and Catalina finds the following statement from Logitech:

Using macOS 10.15 Catalina?

As of macOS 10.15 Catalina, 32-bit software such as MyHarmony and Logitech Harmony Remote legacy software will no longer run and will require an update for it to work. If you’re using a hub-based remote, you will need to use the Harmony app on your iOS or Android device. You can also continue to use MyHarmony on a computer running Windows or macOS 10.14 or lower. If you have a remote that doesn't use a hub, you may download the latest software using the links on this page.

We are currently working on a solution for Logitech Harmony Remote legacy software users and anticipate an update by mid-November. In the meantime, you may use macOS 10.14 (or lower) or Windows to update and configure your account.

See Harmony and macOS Catalina for the latest information.

If you should have questions on this or other Logitech products, check with Logitech; the product vendor.

"...anticipate an update by mid-November." Which, if you noticed, has come and gone.

If Logitech hasn't published new software by December 1st, I'd write off the remote, and maybe Logitech too. I'm in the exact situation as you are, and it seems to me that Logitech may have decided that forcing its customers to buy new devices is more advantageous to their bottom line than updating software for old devices.