I like your views on this text. It is certainly true that Jesus would reach out to those with whom organized religion had little to do. That is true, in fact, throughout the Bible. God tended to choose those for whom others had little regard such as Moses, David, Rahab, Joseph, and Saul. The list goes on and on. That's a great point.
It's also a good point that He did not leave them as He found them. To stay in the presence of Christ was to turn away from sin and move towards righteousness. It certainly happened with Zacchaeus, and still happens today. It can be argued that it is the identifying mark of genuine Christianity.
"Come on down, dude, party's at your place." (That's essentially what the Greek text translates to in modern jargon.)
Not sure how you get that. The Greek simply says that Jesus will stay at his house. The likelihood is he is there to have a meal, but even that's not directly stated. But at any rate, it's not terribly important. Your overview is still quite good.