I think he does. I think that message has been watered down over the past few decades, but it is still there, and as has recently been discussed and in this article, the Popes have been calling us to evangelize.
It is a scary thing to admit and to realize, but I was bound fore hell before I became Catholic. I had the morals of an ally cat. I knew it was wrong, but I lacked the graces I needed to put such things aside. Those graces did come however when I became Catholic.
The fact that I became Catholic was not enough, I had to live Catholic. We are all still in danger so long as we walk this earth. Is salvation possible outside of the Church? I believe it to be, but the odds are much worse outside the church, especially for former alley cats like me.
I know the CC has tools that aren’t available outside the church, chiefly the sacraments, that makes my fight much easier. Those waiting for a deathbed conversion should listen to what the Saints have said about it. While deathbed conversions do happen, for me it took time, even after my conversion to get a fighting chance to resist temptation. It doesn’t always happen “overnight,” and the battle will not be won or lost until it is time for me to leave earth.
God does forgive me when I ask him to, but being a sinner, my natural state is to be in sin. I have to show overall progress, and I have to take full advantage of the time I have.
So yes, for most or all of us, God does indeed wants us to be Catholic.