How to readd a macbook into Apple school manager after releasing the device from ASM?

You are correct in your findings. To side load using Apple Configurator, the Mac must be an x86 T2 or ARM Mac. Basically, this is mostly the 2018 Mac models and newer (Macs with T2). With the 2016, you are out of luck. The Apple Configurator process is incredibly easy but it does require compatible devices (T2 or better Macs). And as noted, the side loading process requires the device to be erased in order to trigger manual association.

However, since you are education, you might want to try reaching out to your EDU rep and ask if they could resubmit the device. Resellers are able to do this so Apple EDU should be able to as well. After all, the EDU market is the ultimate example of chain of custody since only Apple and your institution can exist in the hardware linkage.

However, since this is a 2016 model and you may not care. Remember, the power of being in ASM is mostly for automated enrollment. If the device is still in your MDM, then releasing from ASM will have no impact until the device is reset. If you are not reassigning or resetting the unit, then keep managing it MDM until you can decommission it. If you have already reset the unit, your only hope will be with your EDU rep as Apple is the reseller and thus has the power to re-associate the device.

Hope this is helpful.

I've never tried to do it, but theoretically, you could use Apple Configurator for iPhone to do this. You would need to wipe the Mac and start the setup process all over first.