To be honest, I'm more inclined to the majority of round shields on the side of the English medium infantry. My reasons being:
a) easier production - every village had to be able to produce the shields, it's not like you can go to buy twenty shields in the supermarket...
b) not-just-shieldwall - I mean, we are considering royal fyrd but what about the use in village/militialike fyrd? There might be plenty of use in small local skirmishes, bandit raiding and so on. Not always you are in the shieldwall and as mentioned, round, boss-gripped shield is much more versatile and user-friendly for active use.
But to be absolutely honest, I can see the kite shields on armoured thegns and houskarles. At least at some of these.