I believe you are misreading the intent of the sermon. For some historical context, here is an excellent essay

When Jonathan Edwards preached during July, twelve slaves had already been burned and nine were hanged, and the minister had no way of knowing how the horror would end.

So he was describing what God would do to us, were it not for grace, in the very same terms that were being done to real people nearby.

Jonathan Edwards did not create terrifying visions of torture in order to hurl his people into despair. The congregation, unwilling to accept any responsibility for slavery and its trade, needed "Sinners in the Hands of an Angry God" to ease the intolerable pangs of conscience that were provoked by the events in New York.

There was real evil afoot in America at the time, and Edwards was taking a necessary step towards ridding the country of it.

Some do, some don't. Dividing lines aren't completely cut-and-dry, but it is a controversial question among reformed folk. Generally, you'll find "yeses" among cage-stage Calvinists, and also among more confessional Presbyterians, such as those in the ARP or RPCNA, thought it's probably a minority view even in those churches. You'll find more "nos" in mainline and evangelical churches, as well as the less confessional or more baptistic churches. "Yes" is definitely a minority view in general. In the following I will endeavor to provide quotes that accurately represent both sides, including the theological or Biblical basis for their answer.

Yes

Paul Washer says in "The Cross of Christ" (see a related question):

It is not an exaggeration to say that the last thing that the accursed sinner should and will hear when he takes his first step into hell is all of creation standing to its feet and applauding God because He has rid the earth of him. Such is the vileness of those who break God’s law, and such is the disdain of the holy towards the unholy.

C. Matthew McMahon says on page 349 of The Two Wills of God:

The saints should delight in the reprobation of the wicked, thought that be a most difficult statement to make. Augustine, as a result of Paul's exquisite explanations of election and reprobation in Romans 9, came to the same conclusion. ... We come to understand and praise God concerning the damnation of other people. We understand that we could have been what they are. We contemplate their eternal destiny, and bow before the throne to praise the Creator and the Father we have. How awesome is that grace which He bestowed upon us in His Son!

McMahon's reference to Augustine is this quote from Against Two Letters of the Pelagians, Book IV, chapter 16:

And hence, let the vessels of mercy understand how freely mercy is afforded to them, because to the vessels of wrath with whom they have common cause and measure of perdition, is repaid wrath, righteous and due.

No

The great reformed theologian Herman Bavinck says in Reformed Dogmatics:

God is removed from all wickedness and does not will sin and punishment as such and for its own sake nor delight as such in reprobation.

Wayne Grudem, a reformed baptist, says in his Systematic Theology:

Reprobation is viewed as something that brings God sorrow, not delight (see Ezek. 33:11). ... The sorrow of God at the death of the wicked ... helps us understand how appropriate it was that Paul himself felt great sorrow when he thought about the unbelieving Jews who had rejected Christ [referring to Romans 9:1-4].

John Piper says in his sermon Palm Sunday Tears of Sovereign Mercy:

I appeal to you here: pray that God would give you tears. There is so much pain in the world. So much suffering far from you and near you. Pray that God would help you be tenderly moved. When you die and stand before the Judge, Jesus Christ, and he asks you, "How did you feel about the suffering around you?" what will you say? I promise you, you will not feel good about saying, "I saw through to how a lot of people brought their suffering upon themselves by sin or foolishness." You know what I think the Lord will say to that? I think he will say, "I didn't ask you what you saw through. I asked you what you felt?" Jesus felt enough compassion for Jerusalem to weep. If you haven’t shed any tears for somebody's losses but your own, it probably means you’re pretty wrapped up in yourself. So let's repent of our hardness and ask God to give us a heart that is tenderly moved.

Herman Hoeksema says in "The Place of Reprobation in the Preaching of the Gospel":

God does not desire the destruction of the reprobate in the same way in which He delights in the salvation and glory of His chosen people. ... It is also evident that, when preaching on election and reprobation, we must not place them dualistically over against each other. They are not on the same level. They are not corresponding halves of the same thing, but together they form a unity. Reprobation should always be presented as subordinate to election, as serving the latter according to God's counsel. From this it follows that reprobation should not be preached with a certain delight in the doctrine. He who is forever preaching reprobation shows not only that he is harsh and cruel, but also that he has not understood the work of the Lord God. God's love remains the central thought. He has chosen in His eternal love; and, for the sake of this love, He has also reprobated. Thus all God's work becomes a beautiful organic unity. In this way He is and remains God, and He alone. Thus, at the conclusion of all this, we exclaim in adoration with the apostle, "Oh, the depth of the riches both of the wisdom and knowledge of God; for of him and through him and to him are all things! To him be glory forever!"

My Thesis

While you won't find many modern Calvinists preaching firebrand sermons of this sort, that has more to do with the change in American culture than with a change in theology. The point of the sermon is not to rejoice in the suffering of sinners, but to warn of the very real danger (under Calvin's theology) of falling into hell. Edwards was warning against a false sense of security based on striving for holiness rather than relying on the grace of God.

The Great Awakening

It's hard to emphasize the difference between the way sermons from the First Awakening are viewed now compared to the way they were viewed at the time. For one thing, unlike later revival movements, the target of Edwards' and Whitefield's sermons were very devote, conservative, religious, churchgoing Christians. In other words, when Edwards says, "he looks upon you as worthy of nothing else, but to be cast into the fire" we need to picture folks who resemble Westboro Baptist Church¹, not the people they attack.

Edwards was actually addressing a church that fit into the stereotype of intolerant Puritanism:

How dreadful is the state of those that are daily and hourly in the danger of this great wrath and infinite misery! But this is the dismal case of every soul in this congregation that has not been born again, however moral and strict, sober and religious, they may otherwise be. Oh that you would consider it, whether you be young or old! There is reason to think, that there are many in this congregation now hearing this discourse, that will actually be the subjects of this very misery to all eternity. We know not who they are, or in what seats they sit, or what thoughts they now have. It may be they are now at ease, and hear all these things without much disturbance, and are now flattering themselves that they are not the persons, promising themselves that they shall escape.

The picture Edwards paints is not a crowd of Christians rejoicing over the destruction of evil, but a congregation that feels pretty self-righteous and confident that hell is the destination of other people. The Great Awakening has sometimes been credited with breaking up the conservative hold on American and fueling (in part) the American Revolution.

The Danger of Hell

Calvinists tend to be more sympathetic to this sort of firebrand preaching since they believe a literal hell exists to punish sinners. Warning people about the danger of hell is actually loving in this context. If you happen to know that there's a speed trap ahead, it's a kindness to tell the driver that they might want to slow down. At least part of the revulsion to hell-fire and damnation sermons comes from folks who don't believe hell exists or that sin carries any eternal consequences.²

Further, Calvinists are especially known for believing, along with Paul, that:

For there is no distinction: for all have sinned and fall short of the glory of God,—Romans 3:22-23 (ESV)

and:

But because of your hard and impenitent heart you are storing up wrath for yourself on the day of wrath when God's righteous judgment will be revealed.—Romans 2:5 (ESV)

A Gracious Solution

In our eagerness to read the shocking bits of sermons such as "Sinners in the Hands of an Angry God", we skip over or forget impassioned pleas for sinners to find salvation:

And now you have an extraordinary opportunity, a day wherein Christ has thrown the door of mercy wide open, and stands in calling and crying with a loud voice to poor sinners; a day wherein many are flocking to him, and pressing into the kingdom of God. Many are daily coming from the east, west, north and south; many that were very lately in the same miserable condition that you are in, are now in a happy state, with their hearts filled with love to him who has loved them, and washed them from their sins in his own blood, and rejoicing in hope of the glory of God. How awful is it to be left behind at such a day! To see so many others feasting, while you are pining and perishing! To see so many rejoicing and singing for joy of heart, while you have cause to mourn for sorrow of heart, and howl for vexation of spirit! How can you rest one moment in such a condition? Are not your souls as precious as the souls of the people at Suffield, where they are flocking from day to day to Christ?

The point wasn't to enjoy the suffering of others, but to wake people up to the problem and point them to the solution. Whether or not we agree that the problem exists or agree with the proposed solution, we must not assume that Calvinists are pleased to see others suffer anymore than God is:

But if a wicked person turns away from all his sins that he has committed and keeps all my statutes and does what is just and right, he shall surely live; he shall not die. None of the transgressions that he has committed shall be remembered against him; for the righteousness that he has done he shall live. Have I any pleasure in the death of the wicked, declares the Lord GOD, and not rather that he should turn from his way and live? But when a righteous person turns away from his righteousness and does injustice and does the same abominations that the wicked person does, shall he live? None of the righteous deeds that he has done shall be remembered; for the treachery of which he is guilty and the sin he has committed, for them he shall die.—Ezekiel 18:21-24 (ESV)

Conclusion

Frankly, the idea that Calvinists enjoy talking about the damnation of sinners or feel good about themselves for being better/chosen/more-worthy than others is an unwarranted stereotype. While there are certainly folks who live into that ugly picture, I find most Calvinists to be about the same as other folks in terms of pride, humility, kindness, and cruelty. If anything, Calvinist theology should lead people to be more humble and kind than normal, though evaluating that statement is beyond our scope.

Footnotes:

1. These jerks are by no means the only group giving Calvinism a bad name. They are, however the most "successful" at leading a compliant media to a juicy, but empty, story.

2. I was surprised to learn that the doctrine of universal salvation is among the beliefs circulating in Edwards' time.