"Beginner gains" is a useful abstraction for what I conjecture is an interrelated set of physiological processes. Part of it is the systemic hormonal response to the first time someone does resistance training, part of it is easy improvements to neurological efficiency (i.e. you get better at the movement, so can lift more weight independent of your theoretical maximum muscular strength), part of it is the absence of psychological or physical set points for what constitutes heavy weight. But since the term is just a phrase to refer to what is really multiple complex processes, it's important not to investigate it as its own entity.
If someone starts training and does a lot of upper-body pushing and pulling, they'll get really good at that and their upper body will see improvements in strength, muscle size, and so on. But there will also be system-wide changes caused by that upper-body work: one's metabolism and hormone profile will change, the lower body and trunk will get stronger due to incidental bracing and stabilization work. So the lower body is already seeing improvements without doing any direct work. This is the source of the old adage that an untrained person's bench press can be improved by having them do long-distance cycling--and vice versa. The body adapts specifically to imposed demands, but the imposed demands of cycling include things related to the bench press, and the imposed demands of bench press include things related to the squat.
So when that same upper-body-trained person starts to squat, they will already have the benefit of some "beginner gains", but some other yet-untapped "beginner gains" will be triggered by their new training. They will progress quickly because their legs/posterior chain's neurological efficiency is quite low and ready for quick improvement, and because they may see another systemic response to this new training demand. The effects will not be as dramatic as if they started squatting when they first started training, but I have not seen much evidence that there are any effects of this beginner phase that are not replicable later on.
Regarding linear progression: this technique works in so many situations that it's ridiculous to say that someone should avoid it. It should be the first tool one reaches for when starting a new lift, if only to find the right place to start periodization for that lift.