Logic board replacement or repair (for booting issue)?

I would trust the technician that opened the mac, replaced some parts and knows what they’re doing. Apple training is very good at fault isolation and repair skills and accurate quoting. They would likely have tested a battery or offered you a battery swap which is the typical part failing when the board won’t charge a battery.

The cost of becoming skilled enough to repair a logic board is why it’s almost always cheaper to get a service part from Apple (which you can’t do anymore due to this being vintage/obsolete) and having Apple rework the part and/or scrap it for a new logic board when it’s uneconomical to repair your board.

Unless you can get 5 Macs for scrap prices and try swapping and hope you hit the lottery, swapping the board for one that’s good is going to be the only chance you can get a cheaper repair than paying an expert to source your spare parts or perform some diagnosis on what’s failing on your specific board.