Exactly. If they would have done it immediately, it would have been their decision alone. That’s not embarrassing. By doing it only after an outcry, it looks like they are succumbing to pressure, which makes them look much worse.
The British establishment was trying to do as much damage to USA/British relations as possible before she is finally forced out of 10 Downing Street.
However, I think it has all backfired on them, much like the never Trumpers in the USA. All they do is succeed in alienating the ordinary people and make themselves look like a bunch of incompetent malcontents.