> former members of the Empire chose to stay on as part of the Commonwealth
Being in the Commonwealth doesn't mean "staying on" the British Empire - it just means belonging to a very, very loose trade block on which Britain temporarily exercised an outsized influence. Recent developments (like the inability of subsequent UK governments to replace Commonwealth leadership) have shown that even that influence has now gone. At this point the Commonwealth is little more than an administrative construct for trade-related issues.
Being in the Commonwealth doesn't mean "staying on" the British Empire - it just means belonging to a very, very loose trade block on which Britain temporarily exercised an outsized influence. Recent developments (like the inability of subsequent UK governments to replace Commonwealth leadership) have shown that even that influence has now gone. At this point the Commonwealth is little more than an administrative construct for trade-related issues.