Since I still have the machine in question here, and I'm now interested enough to try and get some rough measurements, I've just videoed it with my phone (30fps video) and done some frame counting, both from a cold boot with nothing cached, and also a repeated launch.
Firstly from a cold boot:
It's hard to tell exactly when the keypress registers, but I believe what I'm seeing is the key being pressed, two frames later the hourglass appears, two frames after that the calculator appears. (The TFT screen will likely be adding at least one frame lag, but let's ignore that for now.) So that's somewhere between 166 and 200ms for a cold launch.
If I close the app and repeat, there's now just one frame between keypress and hourglass, and just one more frame between hourglass and the app appearing, so now nearer 100ms.
Looking at the videos my finger is off the key the first time the app appears, but not the second time - though if I made a special effort to release the key as quickly as possible I now think I could probably just about beat it.
I do do still have both computers set up side-by-side (legacy data from an old business), and the keyboard in question was a Microsoft Comfort Curve 2000 (the calculator button wasn't a proper key, it was one of those squidgy extra keys so beloved of multimedia keyboards, so not as fast to operate as a proper key.)
Anyhow, the point (arguably hypberbolic as it may have been) wasn't about reaction time per se, it was about the older calculator app - and by extension much of the rest of the OS - being a much simpler and less bloated piece of software, and running it on faster-than-contemporaneous hardware makes for a sense of immediacy which is sorely lacking in today's world of web apps.
I'd be very interested to know to what that 300ms "threshold of perceptible delay" applies. You might not notice a window taking 300ms to open - but I'd be willing to bet that when you're highlighting text with the mouse or dragging a slider, you'd be very aware of the UI lagging by nearly 1/3 of a second.