Most (but not all) of those old arcade games were about collecting points and maximizing your score, usually in some kind of endless loop that kept getting harder and harder. Think Pinball, Tetris, Space Invaders, etc. Is there a word or short phrase for such games?
"Score attack" comes to mind. It can be used to characterise both a game as a whole and a score-based game mode within a "larger" game that also offers some non-score based (e.g. plot-driven) gameplay options. See Wikipedia and the (admittedly underrepresented) tag on Steam.
Also, simply calling a game an "arcade" or a "classic arcade" would often imply that attaining the highest score is an important part of it.