Regarding app platform, I'm not sure if dependencies you're using are ready yet but if so you should probably move from UWP, which has restricted access to
APIs (and no available APIs for some areas) to WinAppSDK, which allows calls to both WinRT APIs and the full win32 API surface (in other words it's not sandboxed, and you can declare the integrity level it runs at in your manifest).
For react native support for WinAppSDK, see here for their latest status: New vs. Old Architecture · React Native for Windows + macOS. I haven't tried this, no experience with react, but just putting a link out in case it's useful.
In the meantime, for access to HID APIs from UWP, see c# - Can I Use Windows Hid API Calls in UWP (HidD_GetPreparsedData)? - Stack Overflow, which uses a workaround of calling into a native component in the msix bundle from the UWP app. Don't have experience with the library recommend though (it's OSS MIT). Not sure if the Logitech/Steam issues are a solved problem at this point, but if not, might be worth considering.
Note, for another data point, I'm not able to get the remote to connect on Win11 (no steam, but Logitech keyboard), though in my case I believe it may be a USB power related issue. Computer has USB3 and using an adapter for USB-C. Remote prior to plugging in can light up, but once plugged in, no device manager changes, but then the remote lights do not work, even after unplugging - need to pull the batteries to get back to a working state.
Not blocked on that though, I'm using MacOS instead. Feedback for the MacOS app though is that not having scroll bars in the Button Panel is an accessibility issue. 2 finger trackbar scroll on MacBooks is fairly fragile (there's long threads about it, and none of the workarounds worked), so this required plugging in a mouse with a scroll wheel to be able to program the remote. Scroll bars may look worse, but you can likely auto-hide them for a clean UI, with improved accessibility.