An apparatus in one embodiment includes at least one processing device, with the at least one processing device comprising a processor and a memory coupled to the processor. The at least one processing device is configured to monitor performance of respective ones of a plurality of paths for accessing a logical storage device, and responsive to detection of at least one specified condition in the monitored performance relating to at least a subset of the paths, to switch the logical storage device from utilization of a first access protocol to utilization of a second access protocol different than the first access protocol. For example, in some embodiments, the at least one processing device is configured to switch the logical storage device from a SCSI access protocol to an NVMe access protocol, and vice versa, responsive to congestion, errors or other detected performance conditions currently impacting one of the access protocols.