Back after some days.
I decided to follow my own advice, just to see what I'd get. I googled Seattle International airport. Within a minute I got the website, found their destination route map, noted that there are 4 (four) European destinations you can fly to direct from Seattle. The airlines flying those routes and links to their websites were provided. Another click, and within another minute I had a one-way 'lowest' fare deal currently on offer, Seattle-to-European destination. It took me less time than it takes to type this message.
That's it: those are the direct options. That info can serve as a cost baseline - something to compare with other so-called deals, helps you judge how a fare advertized at $xxx or €yyy compares with others in the price hierarchy. Also, you now have links to those airlines, keeping alert for any other promotional deals they offer.
As aribo already pointed out in post #2, any other possibility requires transfers, and transfers can cost time and/or money, depending on where you go. It also adds planning complications. As is usual in these situations, you now face the trade-off decision - trading off raw ticket price against other convenience factors.
One other tip (again, it's 'logical'), more or less an extension of what I already mentioned. If you want info about flights from Seattle, then ask locally -- even in the sense of a departure location forum -- not just at the destination end (here, in Europe). Not many people on this particular branch forum fly from Seattle, so it's all blind search.
A note on ticket price scanner sites. I've used them to test effectiveness at finding cheapest fares. Result: they don't guarantee that you'll find what you're looking for. Usually, I do better myself with variant of above strategy. Although, it depends on the tradeoffs you're willing to make. For example, I'm not interested in half-day layovers in an airport not close to where I want to go.