Can the logic board be repaired

The computer will boot. To the user selection page. It displays vertical alternating blue/light blue bars. After entering password it begins to load, then goes to a solid crisp grey screen. Then it shuts off or reboots to same user screen.

This had been an intermittent problem until now.

Apple says they don't repair "vintage machines" but says it needs a new logic board.

Can this logic board be repaired?

This is a late 2011 MB Pro.

I've tried plugging it into an external monitor and this happens:

On the computers screen the same vertical bars and when I type in login info the screen goes grey but has double images.

The loading bar continues to the point where screen goes black then comes back on with the proper and clear background image and begins to load programs.

This is when the second monitor detects a signal and shows the same thing.

I got as far having everything load then closed a window then the computer just shut down. I don't get this far without plugging in the external monitor.