Well, the problem is, once you factor out BA, then there is no one airline/alliance which is always easiest (for the lowest business class mileage redemptions).

Lately I find I need a different alliance for each trip. (Fortunately, I've been accumulating miles in AA the most, but also enough in UA and DL and BA and now also AS.) I wanted to go to Alesund Norway a year ago, Star Alliance is the only one than get you there (since it's a city not served by any other alliance, as Scandinavian Air has the alliance monopoly on secondary cities in Scandinavia). Then for this spring I wanted to go to Netherlands, and lo and behold because of my early-enough-in-the-spring travel dates I found Delta at the lowest (125k for business class round trip), while neither other alliance had availability at the time I was looking. Then for early next year I wanted to go to South Africa, and only found availability in oneworld (flying Qatar via Doha but using AA miles).

And the example of Scaninavia shows why "Europe" is way too general a question, because different airlines may be better for different countries in Europe.

Keep in mind that if you transfer to an intra-European airline, their short-haul so-called "business class" seats these days can be worse than domestic USA airline's coach seats . So you may (or may not) prefer to plop down wherever a longhaul jet will take you, and use Europe's excellent trains from there. (But that, of course, requires even more award searching, since you're now looking at lots of European destinations, not just a specific one.)