From OUR POV, Your mileage may vary (and it looks like it does) ;-)
The "invisible church" idea is only necessary after there have been organizational splits. With different organizations sort of dealing with each other with uplifted noses and at arms' length, then there is a need for some account of the sort of big old Church and these itty bitty churches. And for Protestants, of course, that account cannot include any serious pre-eminence for the Holy See and those in actual communion therewith.
Our (arrogant - you be the judge) way of dealing with it is to say that there is one Church. So all the Baptized are in it, though some are not such good members or not so far in. (We have to allow for really awful full fledged Catholics and Spirit-filled Protestants -- the first being impossible to deny and the second being eagerly affirmed, at least these days.)
A Protestant recently argued in favor of the KJV and the "textus receptus" that God providentially preserved a stream of texts and translations from error. We make the analogous claim about the Church. Despite the incredible lack of organization and intentionality which, despite its reputation, really characterizes the Catholic Church, despite the dreadful scoundrels and buffoons, despite the worldly cynicism of some clerics and the perversion of others, the Church is, we think, preserved free from error -- where it really counts.
We see this preservation as an unmerited gift from God, as an example of His persistent faithfulness in the face of our remarkable contempt for Him.
The one advantage we might claim for this point of view is that if you want to have access to bona fide sacraments, go to a Catholic Church, whether Roman catholic, Byszantine, Maronite ... any church in communion with the Holy See. And, we think, if you want bona fide, reliable teaching, then enter the conversation with the magisterium. We think you don't have to guess or to sample this or that denomination.
I'd even say, I HAVE said, if your priest is a jerk, don't worry. The sacraments are still real, and you can go to Scripture and the Fathers and the Catechism for reliable teaching.
Another kind of philosophical advantage, or at least feature, is that our doctrine does not gnosticize or Platonize the "real" Church. It is here, warts and all. YOu can't always tell the good and reliable members, you won't know if you have confused wheat with tares in the individuals you encounter. But it's right here, as Moses says about the Torah.
This also was not meant to be persuasive or polemical but rather expository, and I hope it was useful.
I think you did a great job, myself. Yours is the type of explanation that actually draws people to examine Catholic doctrine a bit deeper - I think it works a LOT better than just telling people they are wrong or expecting them to accept that you are right from the outset!
Very nicely done. This is the type of discussion I like to see around here. :-)