A valid point from Headset starter re Saturday mornings.
I have used the LCY-LPL service once on a Saturday morning back May and will do so again in a fortnight (£37.10 single booked in advance).
The inbound load to LCY on that occasion was healthy - the Theatre & Shopping brigade much in evidence.
The northbound load was equally healthy but was mainly ex-pat Liverpudlian's going to Anfield for a match. I would guess you could substitute that for Toffee fans when they're at Goodison on a Saturday.
I would have thought that a Saturday morning run ex LBA could pick-up enough weekend London breakers plus ex-pat Tykes going north to justify it.