I understand where you're coming from, and I'm not disagreeing that the SSPX has become very worldly and is full of problems, but there is one way in which all Trad priests' hands are tied. Let me put it this way:

What would you think about a chapel which scrutinized each potential member, treating them as non-Catholic until proven otherwise? Background checks, signing statements, written abjurations, cross examination by a panel of clerics and/or laypeople, and "guilty until proven innocent" with regards to being a "True Catholic"?

What would you say about such a group to your friends offline and online? I guarantee you would call them a CULT and you wouldn't be far off base!

It's even more than public relations though: what about the status of Trad chapels and Supplied Jurisdiction? Let's put it this way: the Catholic Church has NEVER been an exclusive club where the lukewarm and bad Catholics are barred at the door. If a chapel is to receive "supplied jurisdiction" from the Church to open essentially a "franchise" of the Catholic Church, don't they have to operate under the bylaws and rules of that franchise?

Operating a franchise without permission of the franchise owner is illegal. In the world of religion, opening a branch of the Catholic Church but refusing to follow all her laws is SCHISM. You're starting your own church.

The Catholic Church (which every single Trad chapel wants to be part of, and connected to) has ALWAYS been, from the very beginning, a net full of fishes, which the fisherman sifts through, throwing the good ones into the KEEP bucket and the crap into the slop bucket for the animals. See also: the parable of the Wheat and the Tares, "Many are called but few are chosen", etc.

As soon as you move beyond that, and try to "improve" upon Christ's Church -- even with the best of intentions -- you begin treading the path of every "reformer" (HERETIC!) in history.

No - Not saying to sign contracts or put them through some austere Inquisition.  But growing up Novus Ordo is much more than simply "going to the wrong Mass."  They still need to have the opportunity to see what the Catholic Church really is, the beliefs, the practices, the true culture.  A Protestant convert gets a thorough catechism course and goes through an Abjuration of Heresy before being accepted into the Church.  Novus Ordo people need some similar chance to get past the "Well I didn't know that!?" stage.  A priest is not supposed to give out Holy Communion willy-nilly.  He is actually handing out Our Lord - Incarnate.  No small responsibility, that.  The Novus Ordo person coming, through God's mercy, to the threshold of Tradition, needs to see what it is to be a real Catholic, and know what he is getting into.  By the grace of God he will persevere, as we all hope to.      

Yes, I think the priest needs to interview them, at least informally, and in no uncertain terms "recommend" very strongly one book at a time. He needs to tell them they probably have a lot of catching up to do, while being kind, charitable, etc. But he needs to impress upon them, with gentle force, that they have an obligation to get up-to-speed. He needs to follow up with them every month and make sure they're actually following his advice/commands. If they are of good will, they won't mind. They will want to become good Catholics. If they don't care, then the priest is at least off the hook.
God doesn't condemn anyone to Hell for acts of someone else's Free Will. What God WILL judge is "did the priest do his job?"

Like I said, the priest needs to speak up boldly and defend Catholic morality and doctrine. If someone leaves because of that, then they leave. That's on them.

Indeed, it just takes a single rotten apple to spoil the entire bushel.  Priests should understand the downward gravity of human nature due to Original Sin.  More often, the bad ones spoil the good ones, rather than the good ones making the spoiled ones bad.  Can all the good apples in a bushel somehow unspoil the rotten apple, or does it work the other way around?  Bishop Williamson was good at explaining this, the drag caused by Original Sin, where if you're not making progress in the spiritual life, you're actually going backwards, since we're swimming against the current of Original Sin.

But I guess that the new SSPX leadership wanted to banish the old guard "prophets of doom and gloom", the pessimism of the Williamson era, and open the windows of the SSPX to the Novus Ordo.  How did that work out after Vatican II?  But I guess they were unable to learn the lesson.

But you know what? If they are not ready for the conversion and they leave, at least they will always remember what the truth was because it was taught to them. One day when they are ready, they will come back to the place where they got that truth. If however they get what the SSPX is "teaching" today, nothing but superficial Catholicism, they will be worse off, because they will believe that they know Catholicism and that it is irrelevant. That is exactly what happened to all of my family members and friends that went to Catholic schools. There is no way to convert them now, they think they know the Catholic faith already, and it is irrelevant.