I've been on both sides of the fence with regards to vendor support. In an extreme case at one company I worked with, when I'd report a slight irregularity or outage to my boss, the first question he'd ask is what is the vendor support ticket number. After all, the company was paying for the support, so it was a very high failing if an employee spent any energy in trying to solve an issue.

The problem with this, of course, was that I don't want to open a vendor ticket until I have something to give them to work on. A random program crashed? First I'd blame the programmer or some other issue, not a kernel bug. But that particular manager just wanted to pin everything on the vendors no matter what.

As an aside, since I've been supporting Linux systems for the last 1.5 decades, I've only had to call in OS vendor support twice -- once was to verify what I already new (clock skew due to a bad hardware timer chip), and another time to satisfy an app developer manager who wouldn't take responsibility for his team's code.

Actually, I should add something -- with Red Hat support, you also get access to a whole history of past case logs at access.redhat.com -- these are discoverable through a Google search, but to see the full solution you have to be logged in. And this has actually kept me from needing to call Red Hat on a number of occasions, so I guess this can count as using Red Hat's support indirectly. And their knowledge base is very well written, esp. when explaining the root cause of specific known past issues.

"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.