'fake' parts aren't 'fake' they are just made by someone else than the OEM. Salvaged parts do not count because for every phone you can fix there is one that you can't. The only proper way is to mandate that phone manufacturers supply original parts to those that need them, but because they'd rather sell you a new phone they don't.

One important reason why you want that original part and not a salvaged or non-OEM one for the battery is safety, this is a device that you carry on your person and the batteries carry a lot of punch. But the car industry has shown the way here, I can buy OEM brake discs and pads for my vehicle, I can also buy them from brand name non-OEM suppliers aka the 'aftermarket'. A similar solution would work well for phones, as would some minor design changes to facilitate repair rather than the opposite: design changes that make it harder to repair a device.

The reason the parts are "fake" is because of Apple's stance on repair. There's no technical reason why Apple can't sell parts at a fair price.