You actually will not extract more sugars that way.
It's definitely possible. The problem is that you have nowhere to heat up sparge water in the meantime. That's why people typically just have the wort run through a copper coil and back to the mash tun. Because in the meantime, your HLT is heating up your sparge water.
The way you want to do it here, you won't really have a way to sparge. So while you're recirculation, yes, you will have water constantly running through your grain bed. However, to maximize extraction you need to rinse it with clean water.