it definitely sounds like a problem on the motherboard with a chip called Tristar or U2. The problem is that this chip is draining more power than what you can put into your phone. Therefore replacing the battery doesn't address the issue at hand.
This is definitely a replaceable part, and is a very common problem. go to a repair shop that does micro soldering and they should be able to help you out.
If you want to give it a go yourself ill point you in the right direction on what to do, but it requires a lot of expensive equipment and even more time learning how to do it.
So I wouldn't recommend doing it if you haven't done it before.
A motherboard is basically a PCB/Circuit board that is inside your phone with thousands of microscopic sized components soldered to it. There is no way to see a motherboard problem, only ways to diagnose a motherboard problem. Kinda the same way that you can't see DNA but you can diagnose problems with it.
My biggest issue is why the repair shop you first went to didn't see the problem in the first place (As it only takes 60 seconds to test for) and refer you else where. We get customers coming in all the time asking for battery replacements and only about 15% actually need them, the rest have the U2 Problem mentioned above.
I hope that answers all your questions.