We call it Mannlicher as a proper term to create a picture in the readers mind as to what kind of stock it is... Like a Thumbhole Stock. Whether it has a Schnabel any where along the forend is immaterial. And whoever thought if it first isn't important either when it comes to creating a picture.
Unfortunately, there are thousands of Bench Rest Shooters in the USA that use Remington Actions. So I suppose Remington does some things right or the Remington actions wouldn't be in demand. There is also an aftermarket for Remington Actions for Custom work. You may be confusing a Remington Barrel for a Remington Action, that is not the way it was meant. I would agree that Remington Barrels have much to be desired. Most Custom work done here will not use a Remington Barrel. This is why Douglas and Hart (among others) make barrels for such use.
The purpose of this poll is to somehow convince Remington to get into a mass produced Mannlicher market. For this to happen, a barrel will come with the action and stock. If you prefer to call it a 'full stock' I doubt anyone would argue with you, but here Mannlicher is the proper term.
Finally, it is just as well that you wouldn't buy a Remington Action because I doubt they would be available in the UK for Custom work any way. There have been other domestic and foreign actions that are superior to Remington Actions, but they have mostly been gobbled up. Custom rifles have also been built out of outdated Military actions such as the Springfield, Mauser and some 'cock on close' British actions. All of which are getting harder to find.