OK, I think I have it covered. As it turns out the cheapest way to reach AY+ Silver is to fly... *drumroll* another airline. All booking classes for AA and US now earn 100% mileage (kilometreage?). To reach silver, you'll need 40000km.
In e.g. February, you can buy an AA-ticketed trip to SFO for 653€ like so:
- HEL-JFK-SFO (~10750km)
- SFO-LHR-HEL (~10460km)
or, alternatively, to HNL for 940€:
- HEL-JFK-LAX-HNL (~14680km)
- HNL-PHX-JFK-HEL (~14740km)
You can therefore hit AY Silver by doing 2 SFO runs (I searched for 0-2 nts) for a total price of 1306€. You can slash off a few hundred by positioning to a Scandinavian airport.