You’re quite right, it’s all in the production! Jonny switched to using a simplified modular setup in 2008, but before that he likely used the exact same Analogue Systems RS8000 which he used on the album recording.
The difference is predominantly in the way that the “kick” is processed. As you mention, the studio version is very distorted/saturated and compressed. There’s also some subtle reverb on the full beat as well, which holds it together in a way that’s missing from the live beat if you hear a recording of a performance. However, at the show it is compensated for by the acoustics of the venue.
Jonny also throws in the beeping square wave for live performances, perhaps to make things more interesting in compensation for the lack of extra processing.