Apple Watch Series 4 fails to update WatchOS 8.8.1 to 10.6.1

I was donated Apple Watch Series 4 (GPS, 44mm case Model: A1978) 2018 with WatchOS 8.8.1.

It seems it should run up to WatchOS 10.6.1 but it refuses to update (*):

In Apple Watch Update Settings there is an error: "Can not search updates. When searching updates there was an error. Try again."

In the paired iPhone SE 2017 iOS 15.8.3 Watch.app Update Settings there is an alert "If you want to update Apple Watch you must first update the iPhone. > Open Update settings" but of course that is not possible for that old iPhone which is already running the latest supported iOS.

So it seems I need a newer paired iPhone that can run a newer iOS (how new?) in order to update the WatchOS, right? If yes, would a workaround be to temporarily pair the Apple Watch to a friend's newer iPhone and update it?

Or is there any compelling reason to update from WatchOS 8.8.1 to 10.6.1 anyway? Would WatchOS 10.6.1 work with the old iOS 15.8.3 and is there any extra benefit compared to WatchOS 8.8.1?

(*) I have rebooted both devices and reset the Apple Watch to factory defaults. Both devices are on the same Wi-Fi. I have also tried to disable Bluetooth in the Apple Watch per some recommendation.