From a personal point of view, I do think we should be throwing shards into more PvP events. They're legitimate cards, and only a little less accessible than the weapons. I've got no idea how balanced they are because I've never seen or used them outside Arena and FGs, but if I saw them in action properly I'd probably get a better idea. People always want new cards, but if we don't use the ones we've got because we think of them as unbalanced or don't like them, what's the point of having them at all? Dimensional Shield, one of the most controversial cards in the game (particularly in restricted environments like War) is unrestricted in most events, so I don't see why we shouldn't give shards a chance.
I do have two problems with them being in War, however. Firstly, I really don't like how some shards are still element-biased. For example a Life-legal duo with SoGs would be notably weaker than the same duo made by a different team simply because of the different mark used. SoV, SoD and SoBr suffer the same problem to various extents. There's also the point that SoI and SoSe probably wouldn't be very useful in War at all, although I could very well be proven wrong.
My other problem with shards in War is the fact that they're just about unused in all other PvP events. This is a problem particularly for War because it is THE major PvP event. It's meant to be the ultimate test of planning, teamwork, reading your opponents, player skill, and of course getting the RNG to fall in your favour. Ideally everyone should come into the event with knowledge of Elements' meta appropriate for their experience. If War suddenly has all a large number of new cards (some of which are quite powerful) that are rarely seen anywhere else, it'd be a total mess.
I'm completely in favour of shards being used in more events, but until they are they'll have to stay out of War for better or worse.