May contain traces of nut
I guess that depends on the TV, the ones I have explored only have a TV wake-up time rather than a "make a racket in case you have not woken with the audio" time, or any day/week/weekend options. My JL can be programmed to do what you stated in post 16, but if you are already awake to hear those tweets it is not really an alarm that you require - just a radio on a timeswitch (or an arm's reach to the on/off button).What else is there? TVs of course! The alarm functions are far superior.
However, I refuse to regard a TV as an equivalent alternative to a bedside clock-radio, and I'm not sure how your TV fulfills your stated requirement for an "extra alarm for the odd travel wakeup".