i think probably as Emacdoug said, SUF’s thinking as to the warm up has developed a bit over time. Remember it’s just like any coach who designs a program. part is based on science, part is based on coaching experience. They often are consistent but sometimes one lags the other or vice versa. Some of the older workouts probably throw you right in with a short warmup thinking that you can compensate through intensity (and partially you probably can, but only partially). Later ones, the science understanding has caught up and they change the warmup a bit to be more comprehensive.
My understanding of the science, is that partially it just requires time and there is no way to shortcut a proper warmup. When you begin exercising, your body starts creating these hormones (the catecholamines epinephrine and norepinephrine) which get your body ready. They dilate peripheral blood vessels but also activate lipolysis (i.e. freeing up ATP from stored fats). This is important even for intense exercise that requires glycogen (because think, if even 10% extra energy is available from lipolysis, you can go that much longer / harder with your glycogen stores before they tap out). But they take time to build up and then the lipolysis they stimulate itself takes time. So if you are jumping straight into intensity, including as part of your warmup, you may be jump-starting the catecholamine flow but your’e also just chomping through glycogen without any support yet from the lipids. This is probably also why people say that the fitter you are, the more warmup you need, because your energy demands are higher and so more ATP needs to be made available to support it (i.e. more lipids need to be mobilized in this step).
So i don’t think there’s really any shortcut. Or rather you CAN skip it, but there’s a consequence. But if you can start easy spinning with gradually increasing intensity for even just five to ten minutes before you start the video, this can be a hack that you might have time for. You can even finish your coffee while you spin