01
The game did not appear at all
Open the Steam Families settings and make sure that both accounts are indeed in the same family. Check the specific game page: the publisher may not allow family access. Games with third-party keys, accounts or subscriptions may also not be supported.
Check with the owner to see if the game is set to "Private". This game is not shown to other family members.
Check family composition.
Check game support.
Check privacy and third-party requirements.
02
The game is visible, but does not start
Steam Families distributes available copies of the game among members. If your family has one copy and another person is already playing it, wait until it's released or buy a second one. If there are several copies, compare their number with the number of simultaneously playing.
Also check the regional restrictions of a particular game. The text of the Steam message usually helps differentiate a busy copy from an unsupported share.
Other, unoccupied family library games may remain available.
03
If the reason is unclear
Restart the client, open Family Library again, and take a screenshot of the restriction message. To contact Steam Support, please provide the game name, user accounts, and the exact text of the error, but do not provide passwords or Steam Guard codes.
Write down the name of the game.
Save the exact error text.
Check the official Steam Families help.