Taking cold shower / drinking cold water after / during training

(I am a new-comer in biology SE so sorry if violated any rules)

Effect of cold showers after intense training

I have read the answer to this question, but it seems there are discussions on the validity there so I am still confused. For me,

Benefits: increase heat loss by conduction on the skin surface, prevent heatstroke.

Disadvantages: thermoreceptors in the skin detect a drop in skin temperature, sending nerve impulses to the hypothalamus. Then the heat gain center is activated and causes vasoconstriction, reducing blood flow just under the skin hence reduce heat loss (&dangerous for patients with heart disease like the answer says). This also reduces sweat production by sweat glands.

Won’t this ‘chain’ of physiological actions reduce heat loss (which is needed after exercise), hence trapping more heat inside the body instead of cooling in the long run? Or am I wrong when deducing the disadvantages?