"The message from the company basically is they'd prefer the sysadmin to spend their nights and weekends figuring out problems, instead of making a small payment for support service."

So... if there's a friday night problem, you'd prefer staff to just be hanging, chillin' at home, and then when pressed, say "we created a ticket with redhat, nothing more to do!" ?

Critical breakages should keep people working until stuff is fixed. If you want to also pay a bit extra and involve external support people, that's fine, but it's not a magic bullet that just 'fixes' everything. You've now got to account for time to manage working with external support staff, making sure you can get them in to the affected systems, etc.