I’m the original poster. I’ll add one suggestion that I’ve looked at several times, but never implemented yet.
Iceland Air flies from a few North American airports to Europe. Most (all?) of their itineraries stop in Reykjavik, Iceland on the way over the Atlantic. This makes for a more pleasant Atlantic crossing by breaking it up into 2 segments (I.e. less continuous time crammed into a seat on a plane). They will even let you spend multiple days in Iceland before continuing on to mainland Europe.
With Iceland on my list of desired travels, One of these days I may take advantage of this allowable multi day layover on my way to Europe.
Without Iceland, I usually start my itinerary research with flights into Dublin, which is about the closest city you can get to from northeast USA. Then from Dublin, I look for a lower cost airline on to my final destination on the continent. I’ve done this twice so far. One time I spent a night in Dublin and once I just laid over for a couple hours. This approach also usually costs me less frequent flier miles that a direct flight all the way into the continent somewhere.
Stop in Iceland, for sure. It's unique.
I use a variation of your strategy and it works. I fly from the US West to NYC either in coach or with miles and then take business class to Europe. As with you, I occasionally need to plan an overnight in NYC.
As others have mentioned, I use Google Flights and the FlyerTalk site has good leads on low fares. I have also had good luck with a site called FareBoom. I have purchased a number of lower cost business class tickets from them and all was legit.
BR