I think with many things the benefits and detriments of a system are largely less a matter of direct inferiority but rather specialisation
Pavises and siege shields did not really fall out of favour in crossbowman and archery, even in gunpowder times it was not unhear of for specific units there were not supposed to be mobile / that were using far more powerful armaments, to construct varying degrees of fortifications. They also partially went away because the ease of digging trenches, of pre-made constructs, increased. As well as the ability to mitigate shields as a whole.
The best kind of shield was the armour you were wearing, and the highest quality of it, and the most completed sets, often never had an associated shield, even as cavalry or as a formation fighting knight. Why? Because a shield was often used when one has a deficiency or a weakness to make up for, it is part of the reason you see the kite shield become so popular as chainmail piercing and the weapons arms race develops, and plate settles in; too expensive to be worn in full, but not quite so expensive for a period to not be worn at all. What do you do? You wear plate on your arms and legs, and you don a hauberk, and most importantly, a kite shield. Now you have something that can cover where the plate is not, on top of your hauberk, meaning you functionally haev as much protection as you could want. A kite shield can be great.
But it is extremely inferior to a pavise to most missile infantry, it is less efficient in formations particularly focused on shielded combat, it has much less protection as well, most often, which is important when the purpose of the shield is to easily take volleys of fire, and also to perhaps take strikes by lances, and such. A targe, buckler etc., also grants one benefits, it grants far higher mobility of the shield, far higher versatility and range in disarming, parrying, and so on, often they can be made of much more resilient materials, or far higher thickness, so that they wear out much, much slower than other pieces of equipment, without the cost of a much larger piece.
This argument is equitable to, but even less justifiable than, 'is a rifleman or an armoured fighting vehicle better?', the point is they offer capabilities that the other lacks, as well as with vastly different forms of, and quantities of expense. They are complements of a greater whole, it kind of just loses value in discussion when one is assessing it without bias. Very rarely is something so versatile, and so highly efficient, that it outclasses everything else. A good example is full plate harness, and more sophisticated poleaxes and what not. But even those two items were not without their detriments, they require far more material than pretty much any other piece of equipment, they still have vulnerabilities, they are more complex to train, and demand a higher fitness of the user, they are extremely expensive, etc.
A kite shield is most appropriate for more frugal spending, more specific coverage, and higher mobility in deployment and use, not so much overwhelmingly better, or even significantly better, than anything else.
