Another commenter linked a video which I’ll re-link[0]. Essentially the conclusion of the HSCA was, as you say, literally entirely based on that recording. If the recording hadn’t been there, the HSCA actually was on the verge of concluding the same as Warren. (In other words, if the recording evidence wasn’t there they would’ve concluded Oswald lone).

The recording was, indeed, later found to not even be from the time of the shots but from an entirely different point in time, after police already noticed Kennedy shot. In that very recording, police officers say things in reaction to the shooting. After those voices, the “shots” from which the conclusion was drawn appear. Painfully obviously, that doesn’t make any sense.

Also the recording was in an entirely different point in space too, as it wasn’t even from a bike in the motorcade, as the HSCA thought but it from a different 3-wheeled-motorcycle stationed where Kennedy would’ve been arriving.

Drawing conclusions from that recording about the Kennedy shooting is probably less reliable than drawing them from the Live Aid concert version recording of Bohemian Rhapsody by Queen.

In summary, after adjusting for that lone error in HSCA, Warren and HSCA concluded the same.

[0]: https://www.youtube.com/watch?v=DC8tO16xdrY