So lets see if I have this right... if you shut it down it turns back on no problem. If it goes to sleep it takes 5 days to turn back on.
Just a suggestion... what if you turn off sleep in System Preferences -> Energy Saver? I'm interested if it does the same thing if it stays on but just goes dead.
I would also try a fresh OS install either from a thumb drive or internet recovery in case something in the OS is causing the issue. It's easy and you can reproduce the issue quickly so its something to cross off the list pretty fast.
I assume you tried the obvious SMC, PRAM resets, etc... SMC controls a lot of the hardware so that's the one I would think is most likely.
Once I had a MacBook that had a bad thermal sensor. It would turn itself off and take a few hours to turn back on but that was pretty obvious that it was hot and overheating so I would assume that is not the case here. If it is getting hot then keep that in mind. Also the MacStore replaced the motherboard and that did not fix the issue so they replaced the whole laptop. I am not sure what the actual fix would have been.
CleanMyMac has thermal warnings that can tell you if its overheating. This may also tell you if you have a bad thermal sensor. If you just turned it on and it's reading 100c then there is something wrong.
Let us know if it may be overheating or if just plain going dead works and it's only sleep mode that does this.
Edit: This stood out in an article about overheating...
- Fake MacBook Charger
A normal charger for a MacBook Pro includes three main parts: AC power cord, MagSafe Power Adapter, and MagSafe connector. It’s always good practice to use the original ones that came with your Mac. If you bought one online, it could be fake and might not work well with your MacBook Pro, thereby causing overheating problem and other issues.
How to Fix It: Shop from the Apple Online Store or Local Retailers.
It’s often not so easy to spot a fake MacBook charger, but this YouTube video shares a few awesome tips. Check it out. Also, try to avoid shopping from online marketplaces, other than the official store, for Apple components. Don’t be lured in by lower prices. I can’t stress this enough.