Is there a way to verify that a JobRemoteRun statement executed properly for an asynchronous job? For example, if the system running the primary script is unable to contact the remote agent, I want to be able to modify the script flow based on this.

You can use JobGetStatus to obtain the status of the last job run. You can call it in a loop until it stops returning -2 or -3 (running or executing notification)

: Is there a way to verify that a JobRemoteRun statement executed properly for
: an asynchronous job? For example, if the system running the primary script
: is unable to contact the remote agent, I want to be able to modify the
: script flow based on this.