I believe Capenhurst is valid via Liverpool and Newton-le-Willows due to being no more than 3 miles longer than the shortest route according to the distances given by the
National Rail Timetable:
Capenhurst to Manchester Piccadilly
via Liverpool and Newton-le-Willows: 45 miles (<=46.5 miles, so valid)
via Liverpool and Warrington Central: 47.25 miles (>46.5 miles, so not valid on distance, or by any other means)
via Chester and Newton-le-Willows: 43.5 miles (shortest route, so valid)
via Chester and Stockport: 48.5 miles (>46.5 miles, so not valid on distance, but valid as a mapped route (map MJ))
Capenhurst to Manchester Oxford Road
via Liverpool and Newton-le-Willows: 44.5 miles (<=46 miles, so valid)
via Liverpool and Warrington Central: 46.5 miles (>46 miles, so not valid on distance, or by any other means)
via Chester and Newton-le-Willows: 43 miles (shortest route, so valid)
Capenhurst to Manchester Victoria
via Liverpool and Newton-le-Willows: 44.25 miles (<=45.75 miles, so valid)
via Chester and Newton-le-Willows: 42.75 miles (shortest route, so valid)
The fares from Capenhurst are set by TfW rather than TPE so have different restrictions and can provide better value depending on the ticket required.
Please correct me if I have missed anything.