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.

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)

Rocket. Jump.

Yet, Quake III Arena had no single player. It was a fun MP game, I spent years playing. It's not really the same as Quake and Quake 2.

I think you might be misremembering Doom 3 a bit. Both it and Half Life 2 came out in the same year, and, well... just compare:

https://www.youtube.com/watch?v=aTJ1weGimZQ

and

https://www.youtube.com/watch?v=y_3vMUOayyc

Doom 3's fully real time lighting and bump mapping was technically impressive, and the live interacting UI was very trick, but the character acting and animation was definitely not SOTA. That was Half Life 2. And if we consider impact on the gaming landscape, Doom 3 was if anything a dud. Elements from that game were not taken along, including not even in subsequent Doom games. Meanwhile Half Life 2's approach to storytelling & world building, animations, physics system - those practically defined the next generation.

Doom 3 was SOTA in terms of realtime lighting and shadows, but that's basically it. In terms of visuals, Half-Life 2 with its baked lighting in directional lightmaps (essentially calculating three lightmaps for each surface for lighting coming from different directions, then using those with normal maps during rendering) with radiosity indirect lighting did a much better job with how good environments looked (and it scaled much better than Doom 3 which in lower settings looked worse than Quake 3). Doom 3's character rendering was also subpar compared to Half-Life 2 - let alone character animations mentioned elsewhere (Source/HL2's facial expressions were SOTA for several years after the game was released). Doom 3's physics were also not as complex as HL2's and the game didn't use them much (the expansion did better use of the physics engine and IMO the Grabber feels superior and more seamless in its use compared to the Gravity Gun but the expansion came later and while the Grabber is nice, the rest of the expansion suffered from focusing too much on gimmicks).

In general while Doom 3 has the better (and probably more forward thinking) rendering tech, HL2 also had some very good tech for its time and did a much better use of the tech they had available than Doom 3 did.

That said, personally i enjoy playing Doom 3 much more than HL2 but that is largely because Doom 3 plays more like a traditional shooter with very little scripting / storytelling to get in your way (and the little there is you can ignore it without losing anything) - you just shoot demons, find keycards/PDAs to open doors and that's it for the most part. I often just put it in low volume and play some podcast in the background :-P.

As for Far Cry, the game looked too plastic IMO, i remember playing the game and the characters' muscles had specular reflections :-P.

I feel like the Source engine for Half-Life 2 had some industry shaking physics due to their Havok implementation, which released in the same year. Doom 3 had cool gritty horror looks, but HL2 blew it out of the water SOTA wise, in my opinion.

I have attempted to play Doom 3 on three separate occasions in my life, and each time I gave up before ever having a chance to fire a weapon.

HL2 just on physics wins over Doom 3 any day. It's far superior in every aspect.