In comparing LH, BA, and AF, I think you have to take the airports into account. UA does access others, but they are not convenient for transfers. For transfers, FRA is definitely the easiest, unless you get a T4 to T4 in Heathrow.

For lounge quality in business class, BA is significantly better with its Terrace lounges at Heathrow than LH's Business or Senator lounges or AF. I haven't yet tried any of the new LH F lounges in FRA so can't compare F service yet. That said, UA's Red Carpet club doesn't come close to any of them. UA's international F lounges are about the level of the business lounges of the European airlines, although not so crowded.

FRA also has the best ground connections if you want to take a day or two in Europe. LHR has the Heathrow express but it is very expensive, CDG is just a mess to get around (perhaps it will be better if they ever fix the terminal they broke.