Its a strange one, but the main Wigan North Western to Lime Street line is serviced by Northern and TPE, and not Merseyrail - even though all but two stations on the route fall under the Merseyrail umbrella. The yellow M is almost as legendary as that other golden M!!!
You got the brands mixed up (not difficult!).
Merseytravel is the PTE that controls services in the Merseyside area (and a few places outside), and brands all its stations with the yellow M.
Merseyrail is the TOC for the 3rd rail electric lines (Northern and Wirral) via Liverpool Central - nothing else. They work for Merseytravel.
The Liverpool City Region Mayor (Steve Rotheram) controls Merseytravel (and therefore Merseyrail), plus also transport in surrounding places like Halton, as part of the Liverpool City Region deal.
That also funded the new Liverpool-Runcorn-Chester service (operated by TfW).
Merseytravel is also going to own the new fleet of class 777s for Merseyrail.
It's possible the City Line (not Central Line) services to Wigan/Warrington/Manchester could be devolved to the city region body, or equally it could go to Andy Burnham and the Greater Manchester equivalent.
TfN (Transport for the North, based in Leeds) will also have a role in any transfer or break-up of the Northern franchise.
The risk is that places like Warrington, Preston and Chester might get screwed in this process because they are not in any "City Region" or PTE area.
Merseyrail does pretty well on reliability, and probably will next year too.
After that it will be down to the reliability of the new 777s.