iPhone shuts itself off at around 30% battery

That's the textbook sign of a worn battery. It can't deliver enough power when the phone asks for a burst, the voltage dips and the phone shuts down even though the percentage says there's plenty left. Cold weather makes it a lot worse. Check Settings > Battery > Battery Health & Charging. If maximum capacity is down around 80% or lower, or it says service recommended, a battery replacement is the actual fix. Draining it to zero and recharging to recalibrate won't change anything if the cell itself is tired.