May I ask how you work that out please? More convenient for what? Given that LSP is approx 5 miles away from Liverpool City Centre, not sure is that convenient.
If you live in, say, Formby or Maghull (which probably generate a fair few through business journeys to London), it's surely easier to have one change at South Parkway- which is after all designed as an interchange- and then a same platform connection for the Ormskirk and Kirkby lines than go through to Lime Street, take the Wirral Line one stop in the opposite direction to where you want to go, and change again.
Good to hear about more ticket machines going in- my experience of using Birkenhead North on a more or less monthly basis to visit family is that you invariably end up with about three or four people who just want to go to Liverpool waiting for the person who wants to book a return to Euston several days hence with reservations.