or that no-where uses AV when our own MPs use it to elect leaders
I'm not sure what claim you're referring to here, but surely the intention must have been that it isn't used for parliamentary elections? That's not true either, because it's used in Australia, but with the proviso that Australian law compels every voter to rank every candidate. Also, I believe that only Labour and Liberal Democrats use AV for their leadership election, and the recent Labour leadership election is perhaps not an advert for AV working well in practice.
I'm not sure what claim you're referring to here, but surely the intention must have been that it isn't used for parliamentary elections? That's not true either, because it's used in Australia, but with the proviso that Australian law compels every voter to rank every candidate. Also, I believe that only Labour and Liberal Democrats use AV for their leadership election, and the recent Labour leadership election is perhaps not an advert for AV working well in practice.