>I’ve learnt to save battery by switching my phone’s wifi off whenever I go into the woods or mountain

I feel like this is not true.

Walking in a city it'd be even worse since there would be 100's networks to check to see if it can connect too. Which I don't think matters much.

Bad cellular reception which you'll hit if you are coming and going out of coverage would matter.

Similarly a wifi you can connect to will drain the battery with background apps and if reception sucks also bad.

But zero wifi you can connect to, I can't see mattering.