I don't consider batch sparging a method to save time, just simpler if you have the space in your mash tun. Waiting 10 minutes after stirring gives the sugars time to migrate out of the grains and dissolve. Think about how long it can take to dissolve some sugar in a cup of tea, even hot tea. It isn't instantaneous.
I suppose that is true but every article I read recently said the same thing, that being you lose efficiency but you gain time.
I'm not sure I'd agree with it being more or less simple. I've seen someone fly sparge and it's way more simple than what I do with a batch sparge. But the difference is he has a great steel tiered gravity fed system with nice burners and great kettles all with valves and thermometers. I find his system much easier but the cost is of course much higher.
I have considered a cheap version that could use another cooler for HLT, a homemade wooden stand, some cheap valves and tubes, and some cpvc. In theory it should work and be pretty cheap, and it would be simple, but it's still more equipment.
In my opinion, batch sparging has one immediate advantage in the very minimal equipment requirements. I'm able to pull it off with a cooler with a valve kit, a work bench, a 5g pot (no valve) and a 8g pot (no valve) and a 1 qt Pyrex measuring cup. If I can also speed things up that is a nice advantage too.
But, next time I might give it 10 minutes after the sparge water is added to see if that helps. I agree with you that it should help. But if I start to run into the time it would take to fly sparge, then I need to consider adding to my equipment.