I think you need to ship out the system to someone with the deeper skills and needed parts.
Think of it like doctors,you go to your primary physician for a check up, clearly their skills are diagnostic and deal with simple issues.
You go to a hospital to have your appendix removed when it gets infected. But, its likely they don’t have the deeper skills to handle cancer.
This is where a specialist is needed! Thats the case here!
Where are you located country and nearest city?
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=== Update (05/29/2020) ===
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For reference here one option for a logic board: [https://www.thebookyard.com/product.php?cPath=39_367_482&products_id=16759|13” MacBook Pro - Logic Board, 2.0GHz i5 8GB Apple P/N 661-05073 - Used]
I think you need to ship out the system to someone with the deeper skills and needed parts.
Think of it like doctors,you go to your primary physician for a check up, clearly their skills are diagnostic and deal with simple issues.
You go to a hospital to have your appendix removed when it gets infected. But, its likely they don’t have the deeper skills to handle cancer.
This is where a specialist is needed! Thats the case here!
Where are you located country and nearest city?