Xrays are only good if you think you acutely broke a vertebrae or have a condition where one vertebrae has slipped out of position relative to the rest in the column -- or if there's a concern about cancer. Since this is the minority of back pain causes (most are discs or soft tissue, not bone), an xray will only do you so much good.

Neurosurgeons like to operate, so you already know the bias there. Some orthopods, too.

I think you'll get the most bang for your buck from a primary care sports medicine doctor. These folks don't operate, but they know their functional physical medicine better than most "carpenter" doctors and work much more closely with their PTs and community providers. This sort of doc may not be who your GP refers you to -- and if not, find one on your own and then phone your GP for a specific referral. The chances you'll get a better diagnosis and route to treatment are way higher.