The ATV3/4 gives you remote access while the ATV4 can run automations. Automations are local if the device running them is local. I was using a iPad in the house but it was unreliability so I switched to an ATV4 and, so far, it seems more reliable.
That said, it doesn't sound like that's your issue. Try getting your wife's access sorted out first. Is she using her own iCloud account? Try revoking her access and doing another invite/accept cycle.
Home occasionally wigs out on me but it always clears up with a force-close of the app.
Worst case, you should delete everything (arrow icon in the upper right) and rebuild HK from scratch.