I think it is interesting to consider the difference between BA and EI with regard to OW elites and thrid-party lounges. Particularly amazing given that EI is frequently considered a less-OW friendly airline.
Anyway, EI use third-party lounges in most airports but as an AA PLT and EXP I have always been invited by EI to use these lounges, eg. airport operated lounge at CDG, MUC. They presumably pay the 3-rd party lounge for me to enter. I always assumed that they then bill AA for the cost.
If EI can facilitate OW elites as well as their own elites why can't BA?