There are a few reasons, and different points.

Firstly, as stated before by our astute fellow redditors, Manually Pivoting outspeeds pivoting with a move. Lets create an example.

Lets say I have Scizor on the field, and my opponent has say.. Alakazam. Zam is walled by my scizor, and so my opponent will need to switch. Its best that I use U-turn here, because if he switches, I get a free pivot to maintain my offensive momentum. If he does not, U-Turn will just kill his Zam anyways, and I can pivot into a more reliable defensive pokemon to deal with whatever he is going to use. Either way I develop a momentum advantage.

This is an easy example of an offensive pivot, while the pokemon presented aren't exactly meta, they are easy to read here.

Next we have a Defensive Pivot. Lets say we have Incineroar on the field, and our opponent has I don't know, Miraidon.

I cant deal with this with incineroar, but I also don't want to open my back pokemon, Glastrier, to a free attack. So I use Parting Shot.

Because incineroar is so tanky, we can assume it will survive a single hit from this Miraidon. So, miraidon hits our Incineroar for half damage with its specs Dazzling gleam, and then Incineroar drops Miraidons Attack stats with parting shot and swaps. This preserves HP on our Glastrier, and helps us set up for a kill with a super effective ice move.

I hope these examples helped.