Beit Yoseph Shechita is a totally separate subject from the Beit Yoseph hechsher in Israel.
Sphardishe Shechita is machmir. The Beit Yoseph hechsher is not.
Would Rav Ovadia have also been behind the Rabbanut Hashgacha when he occupied the position? Do you also wonder why people don’t eat it? Reb Moshe didn’t eat Chalav Stam even though he matired, I highly doubt Rav Belsky ate OU-D soft cheese even though he allowed it. I don’t see the difficulty of having a respected posek associated with a kashrus kulah; it happens quite often.