I have a slightly higher expectation that first-line tech support can solve my problem if the problem is "you really should have had a self-service way to do this on your website but I'm sure you have a tool for this".
And if that isn't the case, I've mostly found that contrary to stereotype, many first-line tech support people are not such rote script-followers that they can't deal with skipping most of the script when the problem is obviously on their end and going to need real human intervention.
