Hi fff123,
I had similar problem with my iPhone6 - low benchmark scores, quickly dying battery and sluggish performance. Changing the battery in a non-official service center didn't resolve the issue, despite I payed for a "very good quality" battery. Situation went even worse - my phone did restart a couple of times during the morning alarm being connected to the charger and once it just stuck at 50% of charge after being charged during the whole night! So I went to them again and believe it or not - they said to have cleaned up some physical contacts inside the phone which were responsible for power delivery - and all problems gone! I now have full benchmark scores even at 15% of charge, phone works fast like new and battery holds very well.
Service man said it is quite a common issue when contacts inside the phone are oxidized and can't handle the power delivery well due to the additional resistance. He also said the battery controller inside the phone might also break thus leading to a non-sufficient voltage for the electronics (which wasn't my case). iOS is so smart that it senses the lack of power and throttles the CPU speed down to keep it working and not restarting (however it still may restart on extreme drops of power delivery), which is quite a smart thing if you ask me in the end of the day.
I should also add that my iPhone 6 is 2.5 years old and used very very heavily even in high humidity conditions, so oxidization of contacts wasn't a surprise after all.
So even if battery change doesn't solve the problem my advice would be to ask some competent serviceman to clean all internal phone contacts (there are multiple pin connectors there) from possible oxidization and to check the battery controller.
I hope it will help someone not to ditch his old phone in despair.