1. A computer-implemented method for performing input/output path failovers, at least a portion of the method being performed by a computing device comprising at least one processor, the method comprising: identifying a computing system that is connected to a storage array, wherein: the storage array comprises a plurality of storage processors, and each storage processor in the plurality of storage processors is within a plurality of input/output paths that connect the computing system to the storage array, and the storage array is partitioned into a plurality of logical units; detecting a failure of a first input/output path within the plurality of input/output paths used by the computing system to communicate with the storage array, wherein the first input/output path includes a first storage processor; receiving a communication from the storage array indicating that the first storage processor within the first input/output path is unavailable; for each logical unit in the plurality of logical units that uses an input/output path including the first storage processor: identifying a candidate alternate input/output path that includes the first storage processor from a list of candidate alternate input/output paths; excluding the candidate alternate input/output path that includes the first storage processor as a viable alternate path for a failover attempt based on: the candidate alternate input/output path that includes the first storage processor, and the communication from the storage array indicating that the first storage processor within the first input/output path is unavailable; selecting at least one alternate input/output path from the list of candidate alternate input/output paths that includes a second storage processor and does not include the first storage processor, and reconfiguring the computing system to access the storage array via the alternate input/output path that includes the second storage processor.