I’ve heard that after the Temple was destroyed, there was a ban placed on having weddings with music in Jerusalem. Or at least instrumental music. Does the Religious Zionist world still observe this ban? Is it only the Old City? Why or why not?

In the Talmudic Encyclopedia (entry: חתן וכלה note 181), the beginning of this custom of not using musical instruments at a wedding,(other than drums) began 150 years ago. It is brought in the name of Rav Shlomo Oirbach zt'l, that although there is no written source for this custom other than that Rav Yoseph Chaim Zonenfeld zt"l said so in the name of the "Imrei Bina", those who abide by it do so both in the Old city of Jerusalem as well as the new city. It is not just an issue if the religious Zionists observe this custom or not, because Sefardim also do not hold by it See Yalkut Yosef (Sova Semachot vol. I pgs. 181-183) and it states there that many Ashkenazi Jews also do not hold by it. From what I see today, most people from all sectors , practice otherwise. All the Best