I don't want to be one of those "you're holding it wrong" types but I think there are a few things you could do to minimize your frustrations.
1. I recommend you rename your HomePod minis (Jack's HomePod, Jill's HomePod) to make it easier to keep track of which is which.
2. I've found the faster way to make sure what I want is playing on the right device is to use AirPlay instead of Handoff. Pick what you want on your iPhone first, then AirPlay it to the preferred HomePod. I'm not sure how well "multi-airplay" works if you wanted to do two streams from your iPhone.
3. If you have any extra Apple devices, consider making those the "jukebox" iPhone/iPads
Cheers.
That's all I did and it works for me exactly as you described. I just talked about that I have the feeling, that they don't think their use cases through anymore.
They offer a reduced UI (which I just learned already exists) instead of an already existing UI with more functionality.
I was hoping to be able to just enter the room if my son, hold the iPhone in the direction of the HPM and the iPhone switches to the correct HPM. As it is now, I have to click on the Airplay-Button, click "control other devices" and select the correct HPM (which of course have different names). One wrong click and my daughters music is off and a tantrum is on.

(Which can be really trouble if you just have managed to work around this BS "up next" list and created a list the little princess might deem hearable.)
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Test:
Try to set an the iPhone in this situation correctly:
"Ok, my child: It's already late. You can not hear this audio book until the end, you need to sleep. But you can hear the first 5 chapters and than you can hear your normal music to sleep." (The audio book is different every day, the music is the same. So fixed playlists are not an option.)
- You want to set the iPhone to play this, hand the stream over to the HPM and then leave the room.
- Please check, what really is on the up next list.
- Add-On: After chapter 4 the kid decides that he doesn't like the audio book and you would like to give him the first 3 chapters of another audio book and then his favorite music.
Keep calm.