I teach American literature to high schoolers and I'm not exactly a scrub when it comes to my knowledge of history and religions, I wasn't exactly a history major or theology major either. When teaching about the Puritan literature, it's important to help the kids understand that their beliefs weren't exactly like what they probably hear in church and these beliefs influenced their writing extensively.
I know that the Puritans believed in predestination and the elect and I thought I had a good understanding of the concept having grown up in a church that has similar beliefs in that regard, but while teaching "Sinners in the Hands of and Angry God" I started wondering if they had a different take on it. That sermon is advocating for the congregation to repent and is aimed at the "unconverted", but for someone who believed the saved were chosen by God before the beginning of the world, that seems to be weird to me. Was this talking about missing out on grace in life and not eternal salvation, or am I missing some subtlety?
In my experience, it is very easy to slide from a belief in predestination to hyper-Calvinism, which is an anti-evangelism "let them come" attitude.
However, Calvinists (of whom I would include Puritans) have a strong history of evangelism and missionary work. Hudson Taylor and William Carey were Calvinists. Carey even made missionaries sign a Form of Agreement that said, "we are sure that only those who are ordained to eternal life will believe, and that God alone can add to the church such as shall be saved." Another great Calvinistic evangelist was Charles Spurgeon and of course the great Jonathan Edwards that you mentioned. The difference between these men and hyper-Calvinists "the frozen chosen" is attitude and action. Instead of "let them come," Calvinists like J. Edwards thought of evangelism as a general call that God would use as he wanted. Spurgeon started preaching at 15 on a soapbox in the coal fields.
But these men were the heirs of the Puritans. Even J. Edwards wasn't a Puritan proper. He's usually just credited as the last Puritan. I know less about them, but I would recommend this article: "Puritan Evangelism" by J. I. Packer, which I found here http://www.the-highway.com/articleJune09.html
TL:DR Predestination does not equal anti-evangelism. The apostle Paul, possibly the greatest missionary in Christian history, used the word "predestined" far too much for that to be true.