Politically speaking, the city of Jerusalem remains contentious. There are some religiously liberal Jews, who are usually politically liberal as well, who feel uncomfortable with a Yom Yerushalayim because of the ongoing political conflict surrounding the city. There are also non-Jews who feel uncomfortable about it, as well. In the 1947 United Nations Partition Plan (Resolution 181) felt uncomfortable enough where they created Jerusalem as a separate international regime that is really not under Israeli or Arab rule. That changed when the Jordanian army illegally occupied it, but that's another story. It's so contentious that Israel is the one country that does not have a U.S. embassy located in its capital. While the Israeli government traded land for peace with the Egyptians by giving them back Mount Sinai, there was some land the Israeli government continues to legally hold. Why Israel keeps the Golan Heights is obvious: from a military strategist standpoint, it helps Israel better protect its northern border. Jerusalem provides no geopolitical or economic advantage. Quite the contrary. Given the political headache it causes, it probably would be easier for the Israeli government to give up a unified Jerusalem (although let's be honest, that wouldn't make Palestine formally recognize Israel, let alone allow for peace in the Middle East). The city of Jerusalem is worth the headache because of its non-geopolitical significance. From a historical perspective, it symbolizes a significant step in Jewish statehood and a step towards political independence for the Jewish state. That argument has its limitations since the Jewish state was already created back in 1948. Even so, the historical significance also has limitations without understanding the religious significance of Jerusalem for the Jewish people.
Jerusalem has been considered the eternal capital of the Jewish people for nearly 3,000 years. The significance of the city of Jerusalem goes back to the Bible when it is first mentioned in the Book of Joshua (Joshua 12:10). The city Jerusalem is mentioned well over 100 times in the Hebrew Bible, and when you count references to the city of Salem (שלם) or implied references, that turns into nearly 700 references to Jerusalem. Since the biblical days, Jerusalem has been attacked, captured, besieged, and destroyed on more than one occasion. The Jewish people lost sovereignty over the city when the Romans captured it in 70 C.E. Jews have lived in the city since then, but Jerusalem has had multiple sovereign rulers over it up until 1967.
Even during the years in which the Jewish people were in exile, the Jewish people never forgot about Jerusalem. In Psalms (137:5), King David says that "if I forget you, Jerusalem, may my right hand forget its skill." The vigor of the Jewish people has been and continues to be symbolized by how much we remember and yearn for Jerusalem.
It can also be a vision for the future. While the etymology of the word Jerusalem is debated, the Talmud says that the word Jerusalem can be divided into two parts: ירא (from the infinitive "to see") and שלם (peace, completeness). Jerusalem is about a vision of peace and represents how we approach a more perfected world. This insight can help explain why we still pray for Jerusalem, even though there is a unified Jerusalem under Israeli sovereignty. The reason why a prayer for Jerusalem is still included in our daily liturgy, as well as why we should celebrate Yom Yerushalayim (and arguably recite Hallel), is because we have not received that level of perfection or peace. On the one hand, Yom Yerushalayim represents what we have accomplished. On the other hand, there is still work to be done, and until that work is done, we pray for Jerusalem. Why? Because Jerusalem is not just a city. It is an idea. We keep up the vigor and the hope that one day, the world as a whole can reach spiritual perfection and peace.