Why is it that Rosh Hashanah we make a Bracha Haetz on the Apple in Honey but no Hadamah is made on the vegetables IF THEY ARE both not directly part of the Seudah?
I can only speak of sephardi custom, but there the custom is to make HaEitz on the pomegranate, adamah on something like the beans, and hakol on either the meat items(heart, lung, head). This order can typically be found in any Sephardi Mahzor. The two I am most familiar with are Ish Matzliah and Shulhan Melekhim.
See the desserts question. Common Ashkenazic practice is that fresh fruit warrants its own bracha. But not vegetables.
As brought in the footnote 35 here,
Chazon Ovadyah (p. 98) writes that one shouldn’t make a Bracha upon the cooked vegetables because they’re considered as part of the meal. Halichot Shlomo (1:18), however, writes that one should make a HaAdama upon the gourd and exempt the other vegetables. To avoid all doubt, Nitei Gavriel 29:18 writes that one should make a HaAdama on a banana.
It's interesting to note Rabbi Tzadok's insightful answer detailing an opposite sefardi minhag than the Chazon Ovadyah. Nonetheless, it seems the Halichot Shlomo would tell you to make a bracha on the vegetables which indeed aren't considered a part of the seudah.