Does the fact that it neglects to base it itself on Test::Builder mean that it does not play nicely with the standard testing phase of CPAN style module distributions?

Not being based on Test::Builder makes it difficult to extend or integrate with other CPAN testing modules. See the excellent documentation that comes with Test::Class for more details.

If Test::Unit seemed reasonably close to what I wanted, what is the closest thing that behaves in an appropriately Perlish fashion?

Test::Class. However, don't forget plain ol' Test::More in cases where Test::Class may be overkill.