It is rare that you come by a motherboard that "IS NOT" repairable. There are several Iphones I have received that the customer was told this (even though it's possible).
First, you should explain your issues in as much detail as possible to help experienced techs troubleshoot your problems.
Second, to replace your Logic Board, is replacing your phone (It is your phone). If it comes to that, you are better served finding a good deal on another phone and replace that way, then you could sell yours to offset the price.
Here on IFIXIT, you can find several experienced techs that can do a repair to most anything that offer mail in services and even no charge in some cases where it is not repairable. Of course you would probably prefer a local shop if available and you have done your homework to make sure they are reputable and good at what they do... but you are here now after going to a shop which tells me they are good and correct, it is not repairable...or they are not.
Please update your question to include all of your symptoms and history of the phone.
Best of luck!