>> You need to bridge the TC..
Yes, that sounds right if it means just letting the TC connect to my existing network as a backup drive, not as a network base station. I didn't see any options to do that, probably because TC says to use Airport Utility to set it up, but Airport Utility's focus is on networks. It's default dialog implies it will create a new network, and I was alert enough to know that I didn't want that, but not alert enough to look past the option to "extend a network".
It's kind of a moot point now because after somehow getting TC to work with Time Machine, AirPort Utility kept popping up after restarting my computer, as if it really wanted me to use TC as a router, so I finally found some time when the rest of the familiy would let me disable internet service temporarily, turned off the exiting router, reset TC to factory settings, then let it be the router.
Works fine, and seems to be much faster, perhaps because it automatically switches to the 5GHz channel when it can, where my old router had 2.4 and 5 GHz channels, but required selecting one or the other statically.