The posek takes into account the dfferent opinions and their reasoning together with his hashkafa and the questioner’s personal situation. The Bet HaLevi, for example, would not answer a kashrut question until he had looked at the questioner’s shoes. The Maharshal had five pesakim for a given shaila: one for an ordinary Jew, one for a rich person, one for a poor person, one for a big tzaddik and one for himself.
Your experience is typical of the Lithuanian ramim and roshei yeshivot as opposed to the rav (In Lithuania, as opposed to Hungary, the rosh yeshiva almost never paskened. this was done by the rav/av bet din).
As for “rabbi shopping” really you should have one rav or at least one rav for an area of Halacha (e.g. Shabbat). However, I have heard of cases where even yeshiva men would not ask their roshei yeshivot shailot in particular areas as they were very machmir and even cases where a rav would tell a questioner to ask another rav he knew had a meikal tradition as he realized that the questioner needed a heter but he could not give it as he was bound by his tradition. What is definitely illegitimate is to ask another rav after being given a “no”.