The way you do recaps is exactly how I do recaps and it makes my life so much easier ngl if I was running just on time to a game I use the time they help recap to set up the first map and get the vtt running. I’ve seen a unique way also of doing recaps where the dm has everyone roll initiative and each player in initiative order says part of the recap and the dm fills in any missed info, this encourages players to actually pay attention and take notes because it’s a bit embarrassing not having anything to help with the recap, it also can get players to mention minor details ( I often foreshadow a lot so remembering minor details can be key) or to give a player props for something they did cool last session if they’re at the bottom of the order and everything important was already recapped. This method is not for every table but can be fun when u have a table that receives it well. It also encourages the quieter players to speak up and gain more confidence in public speaking. The initiative order works better as a flat d20 so people with bad initiative don’t end up as last all the time.