It appears that the issue with Kukai on Windows 11 is not a general issue with the operating system, but instead is related to the beacon connection pop up when syncing with objkt.com. The only way to resolve the issue when it does occur is to first clear the dapp (objkt) cache and retry, and if that doesn't work, then clear the wallet (kukai) cache and retry.
The fact that the issue does not appear to be related to Windows 11 and does work fine on mobile and Mac, suggests that the issue is related to the beacon connection pop up, which is specific to Windows 11. It is possible that the issue is related to a compatibility issue between Windows 11 and the beacon connection pop up, which is causing the connection to become stuck.