As an unrelated aside, I've always thought stations like Hunts Cross, Halewood and other stations with a reasonable amount of potential passengers in their reach should have passing loops added? Basically every station along the CLC line gets a good number of trains passing through, but only once an hour stoppers! At least if there were passing loops, stoppers could let the express services pass, to make up for the lack of quadrupling along the line. Once an hour at stations like that, some of them still within Liverpool city boundaries, is an absolute joke!!
A family member of mine lives in Halewood, and I genuinely prefer to take Merseyrail to Hunts Cross and then walk than gamble on relying on a cramped two- or three-car sprinter that, if missed or cancelled, means an hour's wait.
Right, back on topic.
So would this be the Southport line being extended? It's my person opinion that a separate, OLE-powered service would be much preferred. Given the likely lower level of service, dedicating a 2tph (or more) suburban service using OLE 777s would make much more sense to the commuters than having a weird backwards-J shaped line that hooks back onto its own branch lines, and would prevent the operational annoyances (and even incidents) that power change-overs (3rd rail to battery, or to OLE) can cause.
Since 3rd rail expansion is disallowed, and battery power has significant drawbacks to performance and range, kicking Merseyrail up the arse to start using (and changing over to) OLE would help massively. Not to mention, could the batteries even carry a train up to Aintree from Hunts Cross?
Not to mention that, if my aforementioned (and, in my opinion, much-needed) ideas regarding the airport line
(TLDR: reopen Wavertree, Sefton Park and Speke stations, open a Halewood South station, build a curved branch to the airport with an underground station at Hale and the Airport, and shuttle from there to Lime Street) go into place, there would be OLE 777s running that line too, given that the line is already OLE electrified! Plus, a dual-terminus with the Northern line and "Suburban line" makes operational sense for interchanges with the Airport line, the existing CLC City line, existing services via Runcorn and potential future services via the dormant Fiddler's Ferry line.
I definitely don't think extensions down the line, via Southport line services with batteries, would be a bad idea
at first (if they were to, for example, build the line gradually and run services further and further up it as it gets built) but, looking at the services overall, it makes operational sense and sense for the passengers for Southport services to terminate at South Parkway and "Suburban" services to go from there to Aintree via Broadgreen.
Additionally, you mention services to Warrington. Given Merseyrail services have done their best to isolate their network from sharing track with external traffic (the only exception being Hunts Cross and the crossover to get to it from South Parkway), would Merseyrail services via Ditton Junction to Warrington using the basically-disused Fiddler's Ferry line make more sense, given it's grade-separated from all other traffic (except freight traffic to the Garston car terminus, but I have my own crayonista ideas for that, too
)?