The game is pretty good about taking things like that into account. Even aside from that, a CRT generally adds no lag of its own, displaying the scanline as soon as it begins receiving it, assuming a stable set of sync pulses.

The company that made Rock Band, Harmonix Music Systems, has amazing beatmatching technology that has been used in countless games and is even available in the newest versions of Unreal Engine, thanks to being currently owned by Epic Games. They've been developing this technology since the mid 90s.

The main difference between CRTs and other display types, is that the other display types are all largely sample-and-hold - they display the whole frame line by line, hold it for an amount of time, then begin the process again. They generally lack the decay that happens with a CRT, where the image fades by the time the next frame is to be drawn. Some displays can do black frame insertion, but this is a poor substitute. It requires double the frame rate of the intended output, and it reduces effective brightness. I'd like to see an OLED display that could simulate the phosphor decay on a per-pixel basis, to better simulate the scanning process of a CRT.

I agree that Rockband almost certainly does render a whole frame before scanout. So, yes, there's necessary latency there between the input processing and game logic and everything and the output. Higher refresh mitigates that, but that's not really an option on the kinds of gaming machines Rockband runs on; hopefully it's all locked at 60Hz, because that's what TVs run on (unless maybe ew PAL, but ew), and all the machines have a 240p mode, if they're not cool enough to run 480p or better.

But modern displays add unnecessary latency on top of that. HDTV CRTs often did too. Input devices sometimes add latency too. It's really not too awful as long as it's consistent, real musical instruments have latency too (especially pipe organs! but organists manage to figure it out). It's not as much fun if you need to hit the note when it's visibly far from the 'goal' section, or if the judging becomes visible a long time after the note --- shortening the feedback loop is good.

An analog CRT is as close to zero latency as the source (in this example, Rock Band 3) can provide[0].

More-modern displays (obviously including LCD and OLED, but also CRT displays that themselves have a framebuffer, as was also a thing) always add additional latency.

0: It's not like we can just somehow convert an existing closed-source thing like Rock Band 3 into a 240Hz source.

CRTs far exceed 60Hz though. This FW900 for example goes up to 160Hz.