Why does the electrical power of a generator that is close to the short-circuit fault location decrease?
The wording is confusing. It might be better to say, "The electrical power output from an alternator near the fault must be reduced sharply."

The alternator remains synchronous to the grid, and its terminal voltage is related to its speed, as you say. Therefore, in order to keep its current within limits, its power output must be reduced, which means throttling back its prime mover to reduce the power input.

In other words, the alternator can remain connected to the grid, but the amount of power that it contributes to the grid is way less than it can do under normal circumstances.

Alternatively, if the current rises too quickly, circuit breakers will open in order to protect the alternator, disconnecting it from the grid and reducing its contribution to zero.