Yes, Jesus is saying something particular about the rich here, but not that they are particularly sinful. Rather, they are particularly susceptible to trusting in their money and placing it above God (making an idol of it) which gets in the way of their being willing to follow Jesus and put their faith in Him.
This goes along with another famous teaching of Jesus about money: “No one can serve two masters; for either he will hate the one and love the other, or he will be devoted to one and despise the other. You cannot serve God and wealth.” (Matthew 6:24)
The rich generally have a master they are unwilling to give up. It is money. They think they are the master of their money, but their money is their master.
Yet by God's grace, He can even change the heart of the rich to love Him instead of being devoted to money. Praise God for that. It is indeed a miracle. As crazy as a camel going through the eye of a needle.
In the preceding sections, Jesus basically tells the guy, who’d otherwise lived by the commandments, to come back after giving up all his money and stuff. It seems to me that he’s looking for some more concrete action than an internal change of heart. If I were a Christian I’d make sure to die poor at least, just to be safe.
I believe that a key position of the early church's teaching is that an inward change of the heart is not genuine if it does not manifest in outward concrete action. That's how you reconcile, for example, Paul's statements on salvation by grace and not works, with James' demonstration of his faith by his works. The works themselves don't save you, it is the sincere turning of the heart and belief in Jesus Christ. Paul made this point firmly to counter others who were teaching salvation by adherence to Jewish ceremonial law. But if your belief in Christ and turning of your heart does not result in changes in how you live, specifically following the commandments of Jesus, then James would question the sincerity of your conviction.
It is not the concrete action per se, but even if it were, the concrete action would involve a decision that entails a certain "change of heart". In that very act of relinquishing all one's wealth to follow Christ, you would have assented to and demonstrated your faithfulness to the proposition: to follow God is more important than to be wealthy, and if God were to call me to part ways with my wealth to follow God, then I should and would do it. The concrete act makes manifest the very good in question. It demonstrates and attains the temperance and rational order of the soul in question. Faith without deeds is dead.
We all like to think highly of ourselves, but if we were to find ourselves in that situation, how would we react? Most of us would behave exactly like the rich man. Few have the integrity to meet such austere moral standards. Few of us have the humility to admit that we're no better than the rich man.
This is indeed a passage that is easily misinterpreted by many of its readers. A key to interpreting it correctly, is to note whom Christ is addressing when he uses the expression in question. _It isn't the rich man._ The rich man had already left, because he was not prepared to prioritize God over material riches. The majority of people listening were not rich. They were the common folk. The lesson is, as you've made clear IMO, that the greatest good is God, and prioritizing other goods above God is to make a god of lesser goods, and that this does not lead to the ultimate happiness that is what Heaven is by definition, which consists ultimately of unity with the authentic greatest good first and foremost. And indeed, as other verses state, those who prioritize lesser goods will ultimately have neither the greatest good nor lesser goods. Everyday experience tends to agree with this as well. Obsessive pursuit of lesser goods (which are indeed good) often makes those goods inaccessible to us. We lose on all fronts.
And so, in a very real sense, as you also note, there is nothing especially important about the rich man per se. The poor can lust just as much, if not more, for money than the rich, because the poor can more easily fantasize about and project hopes onto what they do not have; the rich man is more likely to have become disillusioned with his riches.
This goes along with another famous teaching of Jesus about money: “No one can serve two masters; for either he will hate the one and love the other, or he will be devoted to one and despise the other. You cannot serve God and wealth.” (Matthew 6:24)
The rich generally have a master they are unwilling to give up. It is money. They think they are the master of their money, but their money is their master.
Yet by God's grace, He can even change the heart of the rich to love Him instead of being devoted to money. Praise God for that. It is indeed a miracle. As crazy as a camel going through the eye of a needle.