The requirement for LAPL and PPL is for the 80nm/150nm Cross Country to have been completed as "Supervised Solo"
That is because the avereage student does not have a licenace and this is the only way it can be legally completed However; there are people, (military pilots etc ) who have met the requirement in a perfectly legal way and not under the supervision of an FI. There is no safety case to make them repeat it just to satisfy some bureauocrat. Certainly the UK did not follow that stance prior to interference from Europe.
I stand corrected re the distance limit on a Brevet de Basse however; that does not mean a cross country flown legally as the holder of a LAPL would not be counted towards the PPL requirements.
How does a FI supervise a licence holder who is responsible in law for all their actions on a flight when they are acting as PIC? The can only oversee it.