You're doing a good job pulling the weeds as you pass, if the bed is weedfree normally, it's just a matter of puling a couple every now and then. It's not a good idea to leave the weeds till they need hoeing or scrabbling around with a trowel, as you don't want to damage the roots etc which is underneath (and my goodness, they have a root system!)
I'm not so sure about the grit, there's no need to suppress weeds that are going to be whipped up as soon as they emerge, also asparagus likes sandy soil - the grit seems to be a bit heavy. Instinctively I wouldn't use it. We give ours a good layer of rotted manure each autumn when we've cut down the ferns.
