Start up problem when trying to update software

So I have got quite a weird situation with my MacBook Air (early 2015) I've done some searching on the internet and haven't found anything that is like mine. I purchased it second-hand at the start of 2016 worked perfectly fine up until around July-August 2016. I tried turning it on and got no response, thinking that it was just completely drained I plugged in my charger and tried again once it was fully charged. Still nothing.

I then took my MacBook to our only Apple Specialist in town and got told that it was my logic board that is the problem and that it will need to be replaced. I decided to just leave it as I couldn't afford to get it fixed at the time. After a few months passed I decided to give it one last try and it turned on!

This is where it gets weird - My MacBook works fine when it starts up, it even restarts and sleeps/wakes perfectly fine but my problem is when I try to update my software it glitches and gets stuck on the update screen then completely dies - it doesn't boot up again, no start up chime or fan. It just sounds like its trying to start the fan but it gets cut off, it keeps on doing that until my battery is dead. Once my battery is completely dead I plug in my charger press the power button and it starts up again.

I'm just curious as to wether anyone else has had this problem? Or wether anyone can give me any insight as to why it won't turn on unless the battery is completely drained then recharged again and why its glitching when I attempt to update my software?

Thanks!