It seems like this is a special case of project management software. If the existing products can't handle incidents then that software should be improved, not new software written. It's the best way to ensure that everybody on the team knows how to use the software when it's most urgently needed.
E.g. would you change your favorite editor to a different one, in case of an incident? Probably not. So why change project management systems?
While you certainly could cobble together incident response workflows in something like Jira, I think it makes more sense to extend the monitoring and paging tooling (in large part due to the reason you mention— familiarity with the tools that you're using as part of that response).
Did we watch a different presentation? ChatOps isn't new. What you're describing is what I would consider an antiquated practice. Nobody wants to go sniffing around a PM tool at 3AM in the morning.
E.g. would you change your favorite editor to a different one, in case of an incident? Probably not. So why change project management systems?