Why doesn't nodetool work on my Cassandra Cluster?
After some tinkering I figured it out; posting here to close the topic and to help others.

All the scripts in the conf/ directory work in concert and there are a number of environment variables you need to redirect if you install in a non-standard place (i.e. a tarball installation).

For each node in our cluster, we're (now) using:
  (1) cassandra-env.sh
  (2) cassandra-rackdc.properties
  (3) cassandra.yaml
  (4) jvm.options

All the other configuration files are pointed back to the conf/ directory from the tarball using an environment variable.

(2) and (3) set the properties of the node, such as its name, turning on gossip, etc:

(1) and (4) are used by the cassandra bash script when it launches -- the script goes looking for additional configuration files.  We've set it so that the cassandra.in.sh is found in the install area; but previously (1) and (4) were not being found, and these are required to activate JMX, port 7199, which nodetool uses to connect to the cluster.