You shouldn’t have to do that. Instead of trying to get one Google Home device to announce something, you can just use a custom action in your routine: + Add action > Try adding your own > Go home. If your Google Home Hub is set as the Play on device in the Configuration section at the bottom of your routine, then this should cause your smart display to reset when that routine runs. That’s what I do for a Video Doorbell v2 workaround, following the same principles @carverofchoice has described (in fact, he initially pointed me in the right direction on this):
If things are already working the way you expect, then maybe it’s not worth mucking about with it. I just thought you might want to be aware of a way to do it that doesn’t involve having one Google Home device audibly talking to another, in case you want to be able to use your Mini in another room or something.
Regarding your most recent question, is that still an issue for you? The way I interpret your description of the problem, it could be broken in a few places.