Do I have to give every player game time in team sports?

Yes, in practice. Under the Statement of Participation (Sports Rules Article 1, 2024, section 2.6):

• Each team is recommended to play all eligible team members in every game, unless a player is injured or has been removed for disciplinary reasons.
• You must tell competition management before each game if any player on the approved tournament roster will not play because of injury or discipline.
• Failing to play all eligible team members in each game leads to penalties, up to and including forfeit and disqualification.

In Unified Sports there is also a meaningful involvement rule: every teammate should contribute, and no player may dominate play.