Jayarathina Madharasan has already answered the question asked. But I noticed there is another sub question to this which is: "I can't immediately think of a reason why ex cathedra statements would be the only ones to result in infallible doctrines"
I would try to put some light on this valid point.
Ex-cathedra means "solemn papal definitions". It is the Charism the Pope "enjoys in virtue of his office" or literally it means "from the chair".
Many confuse charism of Infallibility as impeccability. Some imagine that Pope cannot sin. Infallibility is not the absence of sin nor is it a charism that belongs only to Pope. Indeed, infallibility also belongs to the body of bishops as a whole, when, in doctrinal unity with Pope, they solemnly teach the doctrine as true.
On infallibility, Vatican II explains that
"Although the individual bishops do not enjoy the prerogative of infallibility, they can nevertheless proclaim Christ’s doctrine infallibly. This is so, even when they are dispersed around the world, provided that while maintaining the bond of unity among themselves and with Peter’s successor and while teaching authentically on a matter of faith or morals, they concur in a single viewpoint as the one that must be held conclusively. This authority is even more clearly verified when, gathered together in an ecumenical council, they are teachers and judges of faith and morals for the universal church. Their definition must then be adhered to with submission of faith (Lumen Gentium 35)."
Some argue that how popes can be infallible if some of them lived scandalously. That is why it is infallibility not impeccability. The doctrine don’t say that Pope wont’ sin or give bad example. Some wonder hoe infallibility could exist if some popes disagreed with others. In fact the accurate understanding of infallibility is that which applies only to solemn, official teachings on faith and morals, not on disciplinary decisions or even to unofficial comment on faith and morals. A Popes private theological opinion are not infallible but only that which is solemnly defined is considered infallible teaching.
Some think that infallibility means that popes are given some special grace that allows them to teach positively whatever truth needs to be known. That is not correct either. Infallibility is not a substitute for theological study on the part of Pope.
What infallibility does do is to prevent Pope from solemnly and formally teaching as truth something that is in fact an error.