When I have multiple accounts (read: 4 kids)...
Do I need for each kid a mobile device to access their account on the TV?
Family Link won't let you use a kid account on your own phone, so managing multiple accounts via one phone is not possible.
Are there other solutions?
Not sure but see this reply for a similar question: https://community.stadia.com/t5/Stadia-on-Chromecast/Second-account/m-p/4379#M101