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