Also, the immune system works better hot, aka fever. Cooling the body can be expected to suppress the immune system.
Before you dismiss proverbs (old wives’ tales), consider that they could persist because they’re true.
I was curious, so I looked it up, and according to [1] and [2], that's a natural reaction to help the nose turn cold, dry air into warm, moist air.
"When cold air enters the nose, it stimulates the sensory nerves within the nasal cavity to activate a process via a cholinergic reflex or pathway. It causes the vessels in the nasal cavity to expand and become engorged, leading to congestion and mucus secretion which produces a runny nose. It’s a compensatory mechanism that’s trying to maintain ideal conditions inside the nose by adding humidity and warmth while filtering the air."
1: https://medicine.uq.edu.au/article/2017/10/why-does-your-nos...
2: https://wexnermedical.osu.edu/blog/is-the-cold-weather-causi...
We have evidence-based medicine exactly because even when it would seem logical, biology puts a curve on that and the opposite of the intended effect happens.