This may sound stupid, but it's one of those unresolved questions I've never really cared much about until now.

What is the point of this switch? I don't get a picture (only the expected thin blue/white horizontal line) when it's switched to "Service" on my Zenith.

On my grandmother's RCA, it was set to "Service" for about 30 years and always got a halfway decent picture. Then, all of a sudden, it got that thin white horizontal line at which point we switched it to "Normal" as instructed (supposedly it had a vertical collapse issue which has yet to be fixed).

Anyway, is there a reason for this switch? Better yet, is there a reason why it worked on my grandmother's TV from October 1979 until at least May 2009 on "Service" and suddenly didn't work when I tried to get a picture on the screen during June of this year?

The TV was designed to be run in the NORMAL position.

If grannys RCA ran for 30 years with the switch set for SERVICE, then, either the switch was bad, or the selection was changed when you pulled the back off for some reason.

There is no inbetween here. Normal or Service.

The reason for the switch is to make setup easier.

Usually you turn the color all they way down, set the switch to service and adjust the 3 screen controls to get a very dim white line. It simply aids in quick setup.
It also can give an idea of crt condition, IE, if you cant get a red line at all the red gun may be weak.

These switches get dirty and cause problems like jumpy vertical or flashes in the brightness.
Sometimes they are best bypassed in the normal mode.