That script calls all of my various test scripts. Kind of like Test::Harness, which I should probably switch over to

You will also probably find the command line prove utility useful (which you'll find in the newer versions of Test::Harness).

(except I didn't see, at first glance, how to pass in that server name argument.)

You could always just use an environment variable. Personally I use the notes facility in Module::Build to store any test configuration information.

For some reason, everytime I have run it with Devel Cover, I just get an analysis of the script runtests.pl, not the code that is tested. Any advice would be welcome.

Is the directory containing your test scripts in @INC (which is ignored by default)? Check out the command line options for ways to get around this.