"Do I really need to worry about this" is hard to answer.
If you need to be able to use the MacBook for long periods of time when not plugged in, then yes, you probably do, because the battery that you have is probably not reliable.
If you mostly only use the MacBook when plugged into A/C power, then no, you probably don't need to worry about it at all. It will most likely work just fine for that purpose.
Do keep an eye out to make sure that the battery doesn't start to bulge or anything like that. If it does, immediately remove it from the MacBook.
As an aside: MacBook batteries generally cost somewhere between $80-130, so I would be highly skeptical of the quality of whatever battery you bought for $20 off Amazon. Where by "highly skeptical" I mean "I would not leave it plugged in and unattended when I was not at home." Which is probably me being overly protective, but as the saying goes, "Better safe than accidentally burning down the apartment complex and/or neighborhood."
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.
Best Answer
Yes, iPhone batteries normally can be charged after going dead (especially after only a few days). It can often take several minutes before coming on.
Since your phone hasn't come on yet, it could indicate a hardware problem and you should take it to an authorized Apple store.