Why won’t my phone charge

While it’s possibly the charge circuit, as Akash suggests, I am a little bit concerned about the diagnostic the shop/tech gave you. I mean, what kind of shop would say the issue is the battery, replace it and not even test it to see if the battery charges properly. The other concern I have is the “wiggle” you say the charge port has. Finally, using cheap car chargers can cause a lot of damage to the charge circuit.

My suggestion would be to find a better shop and have them take a proper look at phone. If you want to try things yourself, than here are my recommendations (you’ve already tried a few of these):

1. Always try another Lightning cable and charger, preferably Apple original or MFi certified units.
2. The Lightning cable 8-pin connector should be sitting flush with the housing and completely inserted. If it is not, there could be lint/dust/debris inside the port impeding a proper connection. You can clean it out with fine point tweezers or a dental pick. Just be careful not to damage the pins inside the Lightning Port.
3. Change the battery. The battery is the weakest link in the entire phone and certainly for charger related issues. It is also the easiest thing to replace. Use a battery utility, such as coconutBattery (for Mac) or 3uTools (for Windows) to test the battery prior to removal. Anything less than 70% of design capacity will require replacement.
4. If the battery is in good condition, you can use a USB Ammeter to determine if the phone is really drawing current when it says it's charging. If it isn't drawing current, then I would change the Lightning/Charge Port.
5. If a new battery and charge port still don't resolve the issue, then you either have faulty replacement parts (possible) or a logic board issue. There is an IC (commonly referred to as Tristar) responsible for communicating with the charging device and uncertified chargers can damage it. Unplug the battery and connect a known-good (preferably Apple original or MFi) charger to the phone. A properly working phone will show an Apple logo and boot loop. If the Tristar IC is bad then the phone will not do anything.