Indeed, the operating airline would hand a lounge invitation for third-party lounges, say the BA lounge at BRU. But these outstation lounges are often small and get extremely crowded at times. For example, BA lounge could get very busy at the time of a BA flight departure. Hence, many of these lounges impose a capacity control. Even within an alliance, it happens that a lounge restricts access to holders of another FFP. I have seen cases where the BA lounge did not accept CX or AA Gold simply because they were too full.