Yeah, I'd figure someone living around there (and when I was a kid I actually did have a friend who lived in the farmhouse just up from Bescar Lane, and I bet they never used it either
) would be more likely to go to KGV, Ormskirk Grammar/Cross Hall (now merged as Ormskirk School on the old Cross Hall site) or Burscough Priory for sixth form. Though Winstanley wasn't unheard of.
I'd venture that the main users of New Lane and Bescar Lane (to the extent that they have any) may well actually be school/college traffic to/from Southport and Burscough.
Actually, thinking on I had a sort-of-friend (more an acquaintance) who lived in Hoscar, too! I don't think he used it either