The true successor of ID software is Half-life 1 with its goldsrc engine... but that simply was made by another studio.

HL1 took both the engine and the genre further + continued the modding culture that brough Counter strike and other mods

(Note I know very well that Half life is not an ID software game, it only took the engine that was auper heavily modified / updated- but it my opinion this is the successor)

goldsrc is based off the q1 engine.