"I will be busy tomorrow."
Mean something more along the lines of "Tomorrow is busy", rather than "(I am) busy tomorrow"? At work, if a client asked if we'd be busy tomorrow, I would say "tomorrow is busy" but I, personally, am not working, so I'm not busy.
On another note, would the following work:
Using を to mark the subject of tomorrow, rather than the understood topic of I?
I don't think so, because in Japanese when you are talking it is generally assumed that you're talking about yourself and WA indicates the topic whereas GA indicates the subject. So, " regarding tomorrow I will be busy". The counterpart of it would be when you are making a question since it would refer mainly to the person you are talking to.
Most of the temporal nouns have two or three different readings. The automated TTS always pronounces the more formal versions for the kanji, so the contributors have chosen to write the more common casual versions in hiragana in order to force system to say the desired reading.
So both are usually written 明日, with the conversational form being the あした reading and the polite form being あす (and the even more formal literary form is みょうにち)
One being in hiragana and one being in kanji in the course is mainly just a workaround for a technical limitation