The particles は and が are a good source of confusion because sometimes it seems like they can be used interchangeably at all times. However, this is definitely not the case. Each particle plays its own role and has its own nuances in a sentence, so knowing these different use cases can be very helpful to distinguish between the two.
The particle 「が」
Common use cases
1. Indicating the subject of a sentence
"The microwave is broken."
This is a simple sentence where the subject is the microwave.
2. Introducing a contrasting clause or phrase
"I tried my hardest, but I still failed."
Here, the speaker wants to convey that they failed, even though they worked hard at it. If they had succeeded, there would be no need to contrast.
3. Following up on something
"Speaking of that matter, is it true?"
Here, a topic was already being discussed so the speaker wants to explicitly follow up on this specific topic.
The particle 「は」
Common use cases
1. Indicating a difference or distinction
"Chief Takahashi is very attractive"
Here, the「は」is used to distinguish chief Takahashi from other employees at the company in terms of attractiveness.
This is reflected in the professor's answer, where he says he doesn't watch TV but he likely does other things to pass the time.
2. Indicating the topic of conversation
"Mount Fuji is beautiful."
Here lies the problem, doesn't it. What if the topic of a conversation happens to be the subject of a sentence? Should the speaker use 「は」or「が」? It's rather tricky to explain, but it roughly depends on the information that's being conveyed. If the speaker talks about something that is shared knowledge between the speaker and the listener, then「は」will do. If the speaker wants to bring new information to the listener's attention, then 「が」is needed.
So, the example sentence using「は」implies that the speaker and listener already both agree that Mount Fuji is beautiful.「富士山がきれいだ。」implies that the speaker thinks Mount Fuji is particularly beautiful, in a way that may not be clear to the listener yet.
Another example that might do this nuance more justice is the following:
"Once upon a time, there lived an old man and an old woman."
"One day, the old man went to cut grass in the hills and the old woman went to the river to wash clothes."
The first sentence uses 「が」because it introduces the old man and woman to the listener. In the following sentence, 「は」is used when talking about the old man and woman and their respective activities, because the listener already knows the story is about them. Funny enough, "an old man/woman" in the first sentence also changes to "the old man/woman" in the second sentence in English, perhaps these ideas are related but that's just speculation.
This is reflected in the question asked to professor Yamashita in your example, as he himself (though being the listener) is the topic and subject of the question.
So what's the deal with「を」?
Simply put, the professor could've said 「テレビを見ません。」and it would've been a perfectly fine sentence meaning "I don't watch TV." However, as you also suspected, the explicit「は」hints at the possibility that the professor does other things to pass the time.
I'm not familiar with the Genki series so I'm sorry if I used words or sentence structures that you've not seen yet. I borrowed them from my sources below.
Sources: Particles explanation (Japanese), Particles guide for writers (Japanese), Differences between は and が for writers (Japanese)