From Pg 75:
When you’ve used your free test and run out of checks to spend, you’re done for this turn.
I guess I was reading “this turn” to mean “the Players’ Turn” instead of “your chance to make a test on this go-round of players spending checks”? Since no one has to donate checks to you if you don’t have any available, I was taking that to mean that a player could end up not being able to spend checks for the rest of the Players’ Turn if they were unable to do it once from lack of donation.
Your reading of it definitely seems much more in-line with the general design of the game.
As such, I’d revise my answers to:
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No, players will never carry checks over from the Player Turn into the GM Turn. However, you won’t have more than 1 player losing checks at the end of the Player Turn.
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There’s no reason not to spend or donate checks if you can. Encourage the players to work toward goals or build up materials for future Missions if they’re otherwise out of ideas.