The 'Club' requirement doesn't only apply to just PPL holders with an FI rating, but also to CPL/ATPL holders. The ANO Schedule 8 includes the same proviso for all flying instruction, regardless of the licence held.
No, it doesn't. The only place that the 'flying club' proviso appears in Schedule 8 is under the privileges of a UK PPL (para 2(a)(i) and 2(a)(ii)). The same restriction does not even apply to the holder of a JAA PPL.
OAT is not a 'club' - it doesn't need to be since it does not employ instructors with a UK PPL. Back when PPL training was undertaken at OAT (and prior to JAR), they had a club, called the Oxford Flyers, which anyone undergoing PPL training, or hiring aircraft privately, was obliged to join.