I remember reading somewhere that the basis of the California class design was 'a ship that might have been introduced in TNG's 5th season,' or something like that. So with that info, and the general design of the ship itself inside and out, indicates that it's not all that old. McMahon did make a comment that these support ships have 'always been around,' but I don't think he literally meant the California class specifically, just random support ships.
One problem that cropped up registry-wise is that the USS Rubidoux NCC-12109, with already too low of a registry, was replaced by the brand-new USS Solvang NCC-12101, a registry even lower than the ship it replaced. Luckily this was the only instance of this.
Also, the design of the California class could have it potentially be a ship built during the Lost Era, after the Ambassador class but before the Galaxy family (Nebula, New Orleans, Challenger, Springfield, Cheyenne, Olympic, Galaxy.) Unfortunately it also shares design elements with the Parliament class, which is a post-TNG design, reinforcing the 'TNG 5th season ship' idea more.
Another bit of weirdness was that Mariner referred to a Steamrunner/Saberrunner class ship as an 'old tub,' which makes zero sense since that ship would have been much newer than any TNG-era ship.
Not quite. The Oberths were science vessels, not second-contact support ships.