Why not? Megyek a buszomhoz.
Do Hungarians really not pause at all between sentences and just kind of slur them together like the audio suggests?
Not really. It sounds like she treated it as one sentence with a comma.
Hungarians do use comma splices pretty freely. It wouldn't be uncommon to see it written like vvsey describes, as one sentence with a comma.
Yes. Especially common among elderly, who don't like to get interrupted during their monologues.
Megyek buszomhoz. - The same question "Why not?"
Megyek a buszomhoz. Sofőr vagyok. It is already accepted.