Is Hajj al-Qiran now considered nonexistent due to not bringing the sacrificial animal (hady) from the beginning of ihram, but rather purchasing a sacrifice voucher (ṣak al-hady) from Mecca, and thus there is no longer anything but Hajj al-Ifrad and Hajj al-Tamattu'?

There is no necessary connection between qiran (combining Hajj and Umrah) and leading the sacrificial animal (hady). It is permissible for one performing qiran to do so without leading the hady, and it is permissible to lead the hady for someone who is not performing qiran; it is recommended, not obligatory. Some scholars have made it obligatory for the one performing qiran to lead the hady, but this opinion is weak. The correct view is that all three types of nusuk (rites: tamattu', qiran, ifrad) are permissible, but they differ in their superiority. Qiran is entering into ihram for Hajj and Umrah together, or entering into ihram for Umrah and then including Hajj with it before commencing the Umrah. The actions of Hajj suffice for both his Hajj and Umrah, and a sacrifice is due upon him. If he cannot find one, then he must fast ten days.