Hi Hugo,
Yes, it is entirely possible your replacement battery is defective, and the symptoms you describe would tend to support that theory. The battery is made up of two components, the cell that stores the power, and a tiny amount of electronics called the BMS, or Battery Management System. If your BMS is bad, the phone won't be able to draw any power from the battery which would explain why your phone only comes on when plugged into an external power source.
In addition, there are temperature and voltage sensors in the BMS, and if the iPhone cannot read them, it will reset the phone in an attempt to recover.
So yeah, everything points to a bad battery. If you still have the old battery you could try plugging it in and seeing if it behaves normally; otherwise you should probably contact the vendor where you purchased the battery and see about getting a replacement. If you bought from iFixit, it has a one year warranty and you shouldn't have any trouble getting it replaced under warranty.
Good luck; let us know what happens!
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