mw: To make a long story short, there are two tracks taken. One is that it’s a Nes that involved the Hatzalah of Klal Yisrael, which necessitates Hallel Mid’rabannan (Gemara in Pesachim).
The second is that there is a Chiyuv (possibly Min HaTorah) to say Hallel for a Nes that leads to the Binyan Beis Hamikdash. (There’s a Teshuvah in the Chassam Sofer about this re: Channukah and Purim.) R’ Schachter uses this notion and says that because we have a Havtacha from HKBH that we won’t lose Eretz Yisrael a third time (he quotes a Gemara and a Ramban Al HaTorah for this), then we know that Yom Ha’atzma’ut will eventually lead up to a Binyan Beis Hamikdash. And similar to Purim, where it was a holiday even before we reached the eventual Binyan Beis Hamikdash, so too by Yom Ha’atzma’ut we are obligated to give Shevach and say Hallel.
The Rabbanut felt that avoiding Chillul Shabbos gives them reason to move the day (I think most in the Rabbanut hold of track #2 because of R’ Herzog).