Doesn't power up unless turned on manually

I got 2 5c's both good both was working fine but started not to charge. I put it down to the cold weather that damaged the battery so I put another battery in and it booted straight up and charged fine.

Now, when I remove the battery and have it disconnected and plug the wire in it should normally boot up to the Apple logo and restart. When I plug the working battery in and plug the wire in it should boot up as it's got a charger in but it doesn't but when I turn it on manually it shows it is charging.

I am not sure if I replace the battery that the phone will be able to recover the battery if it dies.

The phones aren't water damaged and one of them is pretty much new,

I am not sure if this is even anything to worry about as it may work when the battery is out of charge but I just think it's unusual as previously I've always been able to plug the charger in with a disconnected battery to see the iPhone try to boot.