I know that some people specifically daven ke-vasikin on Hoshana Rabbah. Is this only because they stayed awake all night and thus want to daven as early as possible, or is there a more direct and symbolic connection between davening ke-vasikin and Hoshana Rabbah?
There are very big benefits of Davening K'vasikin, and often people can't do it for assorted reasons. Hoshana Rabba is a hidden Yom Tov, similar to Purim and Erev Yom Kippur, it is also considered the end of the Gezar Din. Many people want to take advantage of the Vasikin benefits in order to have a Gemar Tov.
Mateh Mosheh 958 notes that there are five mornings where the custom was to pray early: Rosh HaShanah, Yom Kippur, Hoshana Rabbah, Purim and Tish'ah B'Av.
He writes that praying early on Hoshana Rabbah is hinted at by the verse (Eichah 3:23):
.חֲדָשִׁים לַבְּקָרִים רַבָּה אֱמוּנָתֶךָ
They are renewed every morning — Ample is Your grace!
The word רבה is taken as a reference to Hoshana Rabbah, with the verse pointing to a connection between Hoshana Rabbah and the morning.