Can I use an Oyster Card on any National Rail station within the Travel
Card zones?

I would like to get a pre Pay version but there is no reader at my home
station, Grove Park. There are readers at London Bridge when you alight
.. I often go up to Northumberland Park which also have no readers.
What would happen if I scanned the Oyster at London bridge and then
Liverpool Street on the way to Northumberland Park. Get off, do Spurs,
come back and rescan the card when I get back to Liverpool Street on the
way home.

Oyster just seems a sham if the cards can not be used at the majority of
London national rail stations.