Beis Hillel gave as his second reason for reciting the blessing of wine before the Kiddush the rule of Tadir.
1) I am led to believe that everyone supports the rule of Tadir. If so, what would be Beis Shammai's response to Beis Hillel's reason.
2) Since the rule of Tadir is accepted by everyone, why did Beis Hillel use that merely as a second factor to "even the score" with Beis Shammai. Should not that have been their primary reason?
You are asking a very good question. The answer is that while the principle of "Tadir" is certainly accepted by all, its specific application in different cases depends on other factors. We find in a number of places that the item which is Tadir does not come first, but rather some other factor is involved which gives precedence to the non-Tadir item. For example, in Sukah (56a), there is a dispute which comes first -- Tadir or "Chovas ha'Yom" (whether an item which is Tadir comes another item that is connected to the day's particular obligation). Similarly, in Zevachim (91a), there is a question whether something which has more sanctity comes before something which is Tadir. Also, the commentators discuss whether something which is a Torah obligation comes before something which is Tadir but only a rabbinical obligation (Sha'agas Aryeh #22 versus Pnei Yehoshua and Tzelach to Berachos 51b).
Similarly, regarding your question, we see from the dispute in question between Beis Shamai and Beis Hillel that something which is a causative element comes before something which is Tadir. For this reason, Beis Shamai and Beis Hillel are arguing what is causing what in this case, and then only afterwards does Beis Hillel bring extra support for his opinion from the principle of Tadir.