The point of that was to show how much he has grown to care for her. After her mom said to please leave Kate out of it and brought up Natasha’s death, it seems to have really triggered him.
Also, there’s a lot of parallels between Natasha saying “Let me go” and Kate saying “Pull me up”. He holds on to Natasha until she got away from him and because of this, he lets go of Kate even though she wanted to continue fighting because he cares about her too much to lose her, especially so shortly after losing Nat.
Obviously Kate is going to defy his wishes, but I also think it served to show that not only are they becoming good friends, but it’s delving more into where his brain is in dealing with the loss of Nat and showed he didn’t want to lose someone else.