The tests they give just test what he knows/doesn't know, and not why. Then they start running him through material to teach him what he doesn't have mastered.
If you think he is behind simply because the teaching he had the last few years was poor, then this may be enough (but again, the hourly rate is high).
If you think he is behind because of an underlying LD, then knowing exactly what it is is critical to figure out HOW to teach him. The center-type tutoring probably won't help if this is the case.
And I hate to say, but don't put too much trust in what the teacher is telling you. I had teachers tell me for years that my daughter was "a little slow, but fine". She even made As and Bs in school, although they were hard fought battles with long nights stuggling over homework. The teachers told me they didn't see the need for testing (I asked several times), until she TOTALLY TANKED - really bad- on standardized testing the first year she took it. And then they still tried to tell me that they thought "the test wasn't a true reflection of her abilities". I had to tell them that while it might not have been a true reflection of her abilities, it definitely showed there was some sort of problem and I was getting her tested!!! While we like to think teachers will know all there is to know about learning, including learning disabilities, the truth is most regular classroom teachers know some but not much. They are NOT specialists in this area.
I asked for LD testing last year through the school. We got the oh he doesn't need it song and dance. So we are trying again this year. His ped has offered to write a letter to the school saying she recommends he be tested. We have a meeting at the school on Monday to talk to the intervention team. He's getting tested this year. If I can't get the school to do it then we'll just pay for private testing. Our insurance covers some of this cost. Also our doc is really careful about rxing meds. She requires additional testing above and beyond the norm. We are lucky to be in a position right now that we can afford all of this.