Out of curiosity, I changed /etc/apt/sources.list and restarted the client. apt ran, and it updated the packages on the server for this computer:

# echo >> /etc/apt/sources.list
# /etc/init.d/landscape-client restart
 * Stopping landscape-client daemon                                                                                                                     !tail
                                                                                                                                                 [ OK ]
 * Starting the landscape-client daemon                                                                                                          [ OK ] 
root@inst-022:/var/lib/landscape/client# !tail
tail -F /var/log/landscape/{broker,package-changer,package-reporter}.log
(...)
==> /var/log/landscape/broker.log <==
2013-02-27 16:37:33,549 INFO     [MainThread] Broker started with config /etc/landscape/client.conf

==> /var/log/landscape/package-changer.log <==
2013-02-27 16:37:40,079 WARNING  [MainThread] The package-reporter hasn't run yet, exiting.

==> /var/log/landscape/package-reporter.log <==
2013-02-27 16:37:40,955 INFO     [MainThread] apt_update_interval=21600
2013-02-27 16:37:40,956 INFO     [MainThread] force_apt_update=False
2013-02-27 16:37:40,957 INFO     [MainThread] _apt_sources_have_changed() = True
2013-02-27 16:37:40,957 INFO     [MainThread] _apt_update_timeout_expired(21600)=False
2013-02-27 16:37:40,958 INFO     [MainThread] os.path.exists(/var/lib/landscape/client/package/update-stamp) = False
2013-02-27 16:38:01,538 INFO     [MainThread] Queuing message with changes in known packages: 17 installed, 600 available, 10 available upgrades, 0 locked, 0 not installed, 0 not available, 0 not available upgrades, 0 not locked.

==> /var/log/landscape/package-changer.log <==
2013-02-27 16:38:11,221 INFO     [MainThread] Queuing response with change package results to exchange urgently.

==> /var/log/landscape/package-reporter.log <==
2013-02-27 16:38:11,951 INFO     [MainThread] apt_update_interval=21600
2013-02-27 16:38:11,952 INFO     [MainThread] force_apt_update=False
2013-02-27 16:38:11,953 INFO     [MainThread] _apt_sources_have_changed() = True
2013-02-27 16:38:11,953 INFO     [MainThread] _apt_update_timeout_expired(21600)=False
2013-02-27 16:38:11,954 INFO     [MainThread] os.path.exists(/var/lib/landscape/client/package/update-stamp) = True
2013-02-27 16:38:31,626 INFO     [MainThread] Queuing message with changes in known packages: 0 installed, 0 available, 0 available upgrades, 0 locked, 1 not installed, 0 not available, 0 not available upgrades, 0 not locked.

==> /var/log/landscape/broker.log <==
2013-02-27 16:38:33,558 INFO     [MainThread] Starting urgent message exchange with https://inst-025.virtual-maas.com/message-system.
2013-02-27 16:38:34,644 INFO     [PoolThread-twisted.internet.reactor-1] Sent 26799 bytes and received 150 bytes in 1.08s.
2013-02-27 16:38:34,644 INFO     [MainThread] Switching to normal exchange mode.
2013-02-27 16:38:34,646 INFO     [MainThread] Message exchange completed in 1.09s.

Now the UI is showing me, for this computer:
Available: 43911
Installed: 445
Upgrades: 13 (link)
Held: -

And the "13" link takes me to an actual package listing.