>>> many DVD players are not region-specific & those that are can often be made multi-region if you know which buttons to push - but it's not always easy to find this out
Not any longer. At one time all DVD players were manufactured as 'region free', with the region coding only added to them at the very end of the process. That could be undone by a fairly simple process (which often involved nothing more than opening the disc drawer and entering a code via the remote control).
However the movie distribution companies saw that as a breach of the licensing rights to produce DVD players (which required them to all be coded for use in a specific region), So they've forced the manufacturers to make DVD players that are, from the very beginning of the manufacture process, coded for use in one region only.
While there are still a few 'region-free' DVD players available online, it's no longer possible to remove the region coding from a standard DVD player.