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Was it to the “invisible church at Corinth” that Paul wrote two letters directly to?

There is no “invisible church” in the Bible. There is only the local, visible expression of “the church”. Anything beyond that is simply man’s efforts to explain what is firmly in Christ’s hands.

Another way to look at it - Why did Christ institute “the church” on earth? Obviously first is that He might be glorified in/through her.

But second, the Church was to “go, make disciples” - baptizing and teaching... How can any invisible nebulous entity do that?

I think that you might misunderstand. The idea of a distinct "invisible Church" was really developed by Protestants. It's a way of explaining how, despite much doctrinal and practical division, there is still one Church.

Catholics, on the other hand, believe in it, but also know that there is a visible, institutional Church with a hierarchy and set doctrine.

Another way to look at it - Why did Christ institute “the church” on earth? Obviously first is that He might be glorified in/through her.

But second, the Church was to “go, make disciples” - baptizing and teaching... How can any invisible nebulous entity do that?

The way that Catholics understand the church always confounds me. It's like they have never considered the trinity. There is one God in three persons. There is one one church in many churches. The church in Rome is the church. The church in Corinth is the church. But just as the father is not the son, so the church in Rome is not the church in Corinth.

From the Athanasian creed "So likewise the Father is Lord; the Son Lord; and the Holy Ghost Lord. And yet not three Lords; but one Lord. For like as we are compelled by the Christian verity; to acknowledge every Person by himself to be God and Lord; So are we forbidden by the catholic religion; to say, There are three Gods, or three Lords"

The same understanding applies to the church. In Revelation, the seven churches in Asia are listed. Each has a lampstand, with Jesus standing in the middle. The church in Ephesus "tested those who call themselves apostles and are not, and found them to be false". The church in Ephesus did this on it's own. Their own lampstand; their own judgement. When the scriptures say to take a problem to the church, they are not talking about some singular earthly institution that covers all believers, they are talking about the local church, which is the church.