Is it worth fixing?

If I allow the battery on my MacBookAir 2.1 - A1304 - (2334 Core 2 Duo" 2.13 13") to run down I can get through the boot-up session (very slowly) but within a few seconds of use the mouse and keyboard lock, then the screen goes black. Computer is still on based on light on the front of the case. After that- no response. I get one try per day. Will not boot from USB. Mac genius speculated logic board.

How to diagnose?

How to repair ?

Is it a boat anchor?

Thanks JimF