Had a chance to watch / listen to some of the videos, and, yeah... my dream ideal game takes all of that into consideration while allowing for fast and tense combat resolution. I'm not sure if such an ideal is achievable in the mundane world we physically inhabit.SpoilerShow
Yes I generally agree; however centre-grip shields are also used in wall formations (such as Vikings, Romans etc): perhaps not as tight, though there is some debate on how tight the Greek (etc) formations actually were. The aspis advantage is that its well suited to pushing (using your shoulder), while not loosing too much of the offensive capabilities. I would also think it worked better at bracing against cavalry charges (you feel more secure with a shield braced against your shoulder).
Another difference I have noted is whether a centre grip is orientated horizontally or vertically (image of two reconstructions of shields from Hjortspring):
So what is the difference? I havn't seen any suggestions, but it seem most Roman scutum etc have a horizontal grip, this also seem the most common on Celtic shields.
From playing around with mainly round shieds (where you can turn your hand as you please), but also some experience with scutums here is my take:
The vertical grips allow you to do the things discussed by ThegnThand and Roland Warzecha (which are very good if you want to know how to fight with round shields in general. Especially since these techniques relies on rotation in some aspect.
For completion you can also see Skallagrim and scholagladiatoria
These techniques are indeed very 'offensive' and active. But very hard to pull of with a shield with a horizontal centre grip, such as many Celtic shields (though not all) and scutums. So why having a centre grip horizontally? As they mainly appear on oblong shileds (hexagons, oval or rectangular) I think it is because it allows you to push your shoulder against the upper part of the shield in a push, and thus get some of the advanteges of the Aspis (though not as much), while still retaining the possible reach advantage of the centre grip. This might also be the reason behind the curved scutumes, which cannot be used as Theng Thrand and Roland suggest Viking shields to be used (the flatter oval scutums can to some degree). Also horizontal grips lower the shield, allowing better leg protection in a shield wall.
So: holding the centre vertically is when the shield is used very offensively, horisontally more defensive and in formations, and the aspis style strap might be better for very heavy shields (better resting on your body) and for very intense pushing/bracing against cavalry.
Of course there is also the way shields were used on horseback, and here I thing a horizontal grip is far the best for protecting your legs (though again very little have been done on this by re-enactors as working with horses is difficult). Again a round shield can be used "turned" both ways of course combining the two techniques in one shield.
Another thing a strap on the shield can do for you is freeing your hand (or at least part time freeing it), for other purposes, such as reining a horse or shooting a bow, something like this:
I do not know how common this was in the period.