It is arguable that the Switch is running a more demanding operating system in general than X360. But regardless, you're right there is no technical excuse we know of.
Of course it is running a more demanding OS.
But even though that is the case, there is still zero technical reason to omit voice, it is just not demanding on system resources.
There is a use case scenario that Nintendo is focusing on that is more than likely driving this solution (with an app) over an integrated solution (within console). I think Nintendo is betting that consumers would rather just use their phones for all social mechanisms that any solution they'd create in the Switch. You could argue the phone app is a in-house solution, but I think basing it on the customer's phone is the key. It allows you to also use any other social app instead of or potentially, in combination of their app with any form of head wear you prefer (also alleviating the need to support a bunch of other tech).
Well... Nintendo is wrong, it's a hot cumbersome mess.
Who wants to be fumbling around with phones while playing a game? Since when did we have 4 hands?
I'm still in the general camp that an integrated solution would have been the better customer experience, but I'd like to see if Nintendo has some other new features coming that will make this solution more beneficial.
Specifically, the game specific types of functionality the standalone app can provide by separating the needs of the game from the needs of the online / community functionality.
I don't see how a standalone app running on a smartphone could possibly provide any benefits over an integrated solution, nor has anyone provided a compelling argument on why that is the case yet.
Voice communication has been a "thing" for over a century now, it's not a difficult concept.
Would have loved LTE support as well, but that is another discussion/whinge to be had.