I've bought myself a DaySave for next weekend, and I plan to try it to some Thameslink destinations, if I get the chance, to see what happens.
As far as I'm concerned, Southern is a brand operated by Govia Thameslink Railway, and tickets can only be TOC restricted and not brand restricted, so I'm within my rights, but I'm sure no members of staff would see it that way.
I will note that so far I haven't seen the lettering of the rules behind this statement that brand-restricted fares aren't allowed to be set, but I haven't delved into the TSA deeply yet.
The Ticketing and Settlement Agreement defines an
Operator as: "any person, other than the
RSP, which is bound by this Agreement, or part of it", with a
Dedicated Fare being defined as "a
Fare which entitles the
Purchaser to use the trains of a single
Operator only."
The list of operators is in Schedule 1 to the TSA, but the public version of this is outdated as it still includes Abellio ScotRail, Arriva Rail North, Arriva Trains Wales, East Midlands Trains, NXET Trains (National Express Essex Thameside - aka c2c), Stagecoach South Western Trains, and West Coast Trains.
An updated list - correct to 24th May 2022 - can be found in Schedule 24 to the TSA, more commonly known as the National Rail Conditions of Travel, and that lists trading names as well as the legal entity for each TOC.
Whilst there's no provision in the TSA for Schedule 1 to be overridden, the NRCoT
is an inherent part of the contract under which a ticket is sold, which muddies the waters. That said, as Haywain says in
post #51, I would strongly advise against trying to use a Southern DaySave North of London, as GTR are highly unlikely to roll over and cancel a PF or refund any additional tickets purchased, unlike on Gatwick - London.