Guys sorry for the House of Dragon spoilers but the Dance of the Dragons is famously a big reason for why there are no living dragons at the beginning of GOT so if you think it ends well for anyone, quit now
Hightowers are Bannermen to the Tyrells but since the Lord Tyrell is a baby right now and the Hightowers are close to the crown they have more influence now than Bannermen usually would. GOT followed the Great Houses mostly so Bannermen like the Tyrels and Velaryons didn't show up or were skipped.
Also the Hightowers had more prominence during HotD times anyway, because then Oldtown was still:
i) the largest city in Westeros (by GoT King's Landing has surpassed it)
ii) the HQ of the High Septon and the Faith of the Seven, before they were moved the King's Landing by Baelor.