How about giving your symptoms to the professionals here that have no monetary interest in selling you a new machine or the most expensive part in a Mac? There may possibly be something that Apple overlooked.
Did Apple do any diagnostic tests?
UPDATE
All a white screen indicates on this model is that it is not seeing the hard drive.
Use an external USB drive running Sierra or earlier and boot holding down the option key and see if you can get the external drives icon to come up.
Have you ever experienced display artifacts or other malfunctions?
What did you attempt to update to? Are you using the original Apple hard drive? Has it ever been worked on or upgraded? Has the hard drive/IR cable ever been replaced?
There are know issues on the machine and we will attempt to narrow the problem down and identify exactly what the problem is.
Do you have another Mac with a Firewire port so that you can boot using Target Mode?
How long did Apple take to run these tests and outside of telling you that you need a new logic board did they tell you anything else?