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)
Quake was better for multiplayer though. Personally I enjoyed Quake 2 the most. Quake arena was designed for multiplayer but you had to practice with the cheating bots so it was kinda boring.
IIRC, my problem with Quake multiplayer was that one node acted as the server, so when I connected to a friend’s computer via modem, my lag was a serious disadvantage (neither my friend nor I ever won as clients). Doom, on the other hand, simply froze both computers whenever there was a communication issue.
Although, to be fair, we played Doom over an RS-232 cable - hauling a PC across the city every weekend was a testament to our love for the game :)
Quake 2 multiplayer is such a blast. The cat-and-mouse chase fights in that game is what defines the genre of "arena shooter" for me, there's still nothing else really like it.
The campaign has a place in my heart too, even if it's not perfect. A lot of DOOM's level design was predicated on claustrophobic interiors, and when you go "outside" in many levels it feels like a glorified courtyard. From the very first level, Quake 2 pushes hard to create an illusion of environmental complexity that plays very distinct from Quake 1 or DOOM.
Personally I think Unreal Tournament perfected the genre when it comes to multiplayer. Q1 was a lot of spamming of grenades and so on. Q2 was better. Still a lot of chaos. UT99 was also chaos but you could combine it with perfect moves and high precision shots. Great games all of them. I used to be an elite UT99 player but as the pace kept increasing along with my age my reaction time was simple not good enough anymore. Even if tactics compensated a lot those games are not like sneaking around in CS. I mostly played CTF. Good old times.
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)