What a cool-down actually accomplishes
A gradual reduction in intensity (rather than stopping abruptly) helps heart rate and blood pressure return to baseline more comfortably, and may modestly support the beginning of the recovery process.
Static stretching's actual role post-workout
Post-workout, when muscles are warm, is a genuinely good time for static stretching to maintain or improve flexibility — the earlier caution about static stretching applies mainly to pre-workout timing, not after.
Whether stretching actually prevents next-day soreness
Research on stretching's effect on reducing delayed-onset muscle soreness is genuinely mixed and modest at best — worth doing for flexibility and relaxation, but not a guaranteed soreness-prevention tool.
A practical, time-efficient approach
5-10 minutes of light movement followed by stretching the major muscle groups worked that session covers most of the practical benefit without requiring an extensive, time-consuming routine.