The tradition in my group is for a player to do the recap in exchange for an inspiration token (or a hope in our Daggerheart game). Then the DM fills in gaps.
As I’ve started co-DMing the group I’ve also realized that it’s a great way to reinforce the plot points that players may have missed or not seen as important.
One of my groups has the DM giving us some bonus exp for writing a “journal entry”
That’s kinda how Margaret Weiss and Tracy Hickman wrote Dragonlance. One of them DMed while the other took notes. The first trilogy, at least, was a retelling of the campaign they ran in their brand new setting. Which they made, because at that point in D&D there were tons of Dungeons, but scant few Dragons.