If your (their) profiles are truly Scheduled, each such trigger starts, uses and closes a separate instance of SE to run the profile anyway. Nothing the user does* in respect of that failed run will have any effect on the next trigger.
You may also be interested in the option to Silently Fail, on
Modify > Expert > Copy/Delete > 'Silently fail if Destination cannot...'
This will (as per the contextual Help - press F1 with Advanced page open) suppress any Failure result (the Result column value for that profile will blank** instead) and also any log. This has the advantage of suppressing a string of logs with 'What network?' errors (possibly pushing the last valid log 'off the end'), but does mean you have to remember it is in force, and that if you try to run it (even manually) with any Dest-reachability issue (expected or otherwise), it will simply close with no error (because you've told it to). Your choice...
* apart from actively deleting or disabling the Schedule - but that's not a 'response' to an error, that's a different, 'configuration' issue. The point is, there's no concept of 'That failed - wanna stop trying?'
** why 'blank'? It's a compromise. People wanted to avoid that showing Failure if the reason is as per that setting, but leaving it with Success (or whatever the last Result was) implies that was the most-recent situation, which it actually wasn't. Note that the LastRun & LastSuccessfulRun values don't change/update because of any Silent failure/s.
You may also be interested in the option to Silently Fail, on
Modify > Expert > Copy/Delete > 'Silently fail if Destination cannot...'
This will (as per the contextual Help - press F1 with Advanced page open) suppress any Failure result (the Result column value for that profile will blank** instead) and also any log. This has the advantage of suppressing a string of logs with 'What network?' errors (possibly pushing the last valid log 'off the end'), but does mean you have to remember it is in force, and that if you try to run it (even manually) with any Dest-reachability issue (expected or otherwise), it will simply close with no error (because you've told it to). Your choice...
* apart from actively deleting or disabling the Schedule - but that's not a 'response' to an error, that's a different, 'configuration' issue. The point is, there's no concept of 'That failed - wanna stop trying?'
** why 'blank'? It's a compromise. People wanted to avoid that showing Failure if the reason is as per that setting, but leaving it with Success (or whatever the last Result was) implies that was the most-recent situation, which it actually wasn't. Note that the LastRun & LastSuccessfulRun values don't change/update because of any Silent failure/s.