I think that the major reason why they kept Eric Ten Hag was because he was able to win City in the FA cup finals which made the club to qualify for Europa. Even though the club performed poorly through out the season but a trophy was achieved and that brought smiles to the faces of the fans and the players. Some clubs that performed brilliantly was unable to win any trophy which means that he can improve the club performance.
No one knows whether if a new coach is brought into United to replace Ten Hag will perform better and that was why Eric Hag was given his last chance to see it he can improve the club and take them UCL next two season. I will be happy if United can go far in Europa League next season.
To be honest, personally I don't accept that reason as the reason why Manchester United retained Erik Ten Hag, even though maybe that was one of the ingredients they brought when they evaluated the coach's performance. Personally I can't accept one trophy as more valuable than their performance throughout the season, because they can get that one trophy only because they were very lucky in that match.
Let's see what happened in that match, maybe in detail I have also forgotten how the match went because I also saw that match. But let's look at the statistics of the two clubs competing at that time. In terms of play, Manchester City really dominated the game, even at the end they recorded 74% possession of the ball. Possession of the ball is not a benchmark for whether they will win or not, but at least we can see how Manchester United continues to come under pressure.
Let's also compare their meeting record in the last few meetings, we will get a note that Manchester City is superior in their last few meetings. So this is my assessment, even Manchester United being able to win in the final cannot be separated from their luck in that match.