If your phone shuts down with more than 10% charge remaining on the estimate, then you have a voltage / capacity issue in the battery and should have the device tested with Apple diagnostics to check the health of the battery. Think of a small town with a medium sized reservoir and a small water tower. The reservoir level can have water, but if it’s not pressurized (voltage) in the tower, the system can ask for more water (current) than the tower can supply. For a shower, the water just stops, but for an iPhone, it shuts off when the voltage drops too quickly.
The diagnostics read the low battery logs and compare them with thousands of other logs and Apple engineering standards to tell if your battery needs to be replaced.
Technically, those log files are total benefit with no downside. Your phone is out of juice and will shut down anyway. The OS is logging the data so you can take action on it if you choose. They don't cause anything other than logging what led up to the low power condition.
I watch my LowBatteryLog-YYYY-MM-DD-hhmmss.plist log files quite regularly to know when to seek service for my battery.
Technically, the lithium polymer cells can exhibit voltage drops when they are out of specification but not yet failing or in a state where they should be shut off and not used again.
Looking more deeply into your two posted logs, I would focus on these lines primarily:
Awake Time: 03:45:27 (13526)
Standby Time: 04:55:29 (17729)
Partial Charge: 0
Capacity: 0
Voltage: 3469 mV
When Partial Charge is 1 - that means that you plugged it in and the device received a charge between when it was full and first removed from power and it eventually shut itself off due to low power detection. In those cases, I really only focus on the Voltage - knowing when the device decided to preserve the remaining voltage for standby and battery protection.
The log above shows a very short awake time and indicates a likelihood that the battery isn't providing the correct duration and amount of power. Even if the CPU is full use, all radios are on, speaker cranking, brightness max - I expect 4 to 5 hours on most devices.
Unless that 3h45m run was a rare occurrence, my estimate is you need a hardware repair as sleeping more often will allow the sleep time to increase, but never increase the active time.
For newer iOS hardware (think iPhone 5 to 10) the voltage to available current to available capacity can also be due to aging, power management software or hardware issues.
Apple does offer a one year warranty, so if you are the first buyer, you should be able to register and get them to replace the battery for free. I don't know exactly when they discontinued selling that model, but start with trying to register that device to you at support.apple.com using the serial number from the box. You might need to call into support or fax them the receipt since it's so far past when they were expecting to enroll new owners, but once you know if Apple can/will service it, you can decide to repair it through them or on your own.
It's likely the lithium battery has failed permanently due to not being stored for long term with 50% charge. I typically get out my old devices once a year and charge then to 50% and then power it off. If the voltage drops too low then you will need to replace the battery as it will never charge or even hold but a fraction of the intended energy.
The customer care has given you good advice and it's great that they offered you the money back. I would say if you are happy with the price, look at repairing the battery yourself. That design was very easy to self-service - especially if you realize it's unique and will take your time with the service.
The parts should be readily available and the tools are inexpensive. Older phones (original) and newer phones are much harder to replace the battery due to tools or design of the connectors and internals. I'd say keep the phone unless you really can't afford the replacement cost of the battery.
Best Answer
Like you said in a comment, it's more a software limitation. Even though there's probably some hardware safeguard.
If you plug your iPhone a millisecond before it reaches the shutdown limit, then, no harm done, it doesn't need to shut down, power is there to keep the phone running and recharge the battery.
If you plug you phone after the shutdown limit, then the limitation is in place to ensure things will go well the next time you power it up. If it allowed to power up the phone any time, then what if it is unplugged while booting. There wouldn't be enough juice for the phone to be fully back on and it would shutdown while booting.
That's something that could damage the phone or the OS.
So to ensure that this wouldn't happen, Apple probably put the arbitrary 5% limit.