The issue isn't the battery, it will be the IC chip in the phone.
It turns out that if anyone uses a third party charger it may not only cause physical damage, but it can cause internal damage to the U2 IC chip in the logic board.
The U2 IC chip controls the charge to the battery, the sleep/wake button, some USB functions, and regulates the charging power to the power IC that actually charges the phone. When damaged, the chip can fail to work properly, which prevents an iPhone 5 from turning back on. While a fresh replacement battery will power the iPhone, once the battery is depleted, the issue resurfaces.
So I advice to take it to the phone repair shop as this would be a challenging repair.
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