In my experience, everybody turns their professional manner up a notch and uses great economy in radio transmissions. Remember that everybody on the frequency is clear on the concept here, and most are probably turning the words, "there but for the grace of God go I" over in the back of their minds. It's also kind of neat that all of the other pilots on frequency contribute to the assist when they do this. Team effort. What's more, I've never had to say to other aircraft on the freq "keep it short, I'm working an aircraft in distress here." It just happens. Coolest damned thing. Great example of situational awareness provided by all involved...
Although you're correct in your assumption that we want to keep you from having to do things like change frequencies at a time like this, most shops don't have the luxury of moving everybody else to another frequency. So we all (drivers and controllers) just deal with it. If it goes south, it seldom has to do with frequency congestion or controller workload generated by non-emergency aircraft.
Dave
PS Stop saying "confessed." You're availing yourself of a service here. You're in a spot and we're simply a tool that may be able to help. If you end up having a confessor because of this situation, it'll be a guy sitting across a table from you discussing the facts and their impact on your future in aviation some days or weeks later. It won't be the guy who says "RADAR contact."