Learn Turkish: Makinistin treni çalıştıracağı yok. - The train driver has no intention of starting the train.

This is a common fixed structure using the future participle -(y)acak or -(y)ecek, a possessive suffix, and the word yok. Literally, it translates to "his future-starting does not exist." In everyday Turkish, this pattern is used to emphasize that someone shows absolutely no sign, intention, or likelihood of doing an action.

Why is the subject makinistin in the genitive case instead of just being makinist?

Because the verb has been turned into a noun (nominalized) by the -acağı suffix. In Turkish, the subject of a nominalized verb takes the genitive case (here, -in), and the nominalized verb takes the matching possessive suffix (here, -ı). The grammar literally builds the phrase "the train driver's starting."

The verb root looks like çalışmak (to work). How does it mean "starting" here?

You are right! The root is çalışmak (to work). Adding the causative suffix -tır creates çalıştırmak, which literally means "to make (something) work." When applied to vehicles, machines, or engines, çalıştırmak is the standard verb for "to start" or "to turn on."

Since çalıştıracağı acts like a noun here, why does treni take the accusative case?