I still install counter flashing that is cut into the mortar joints.

Is the top of the flashing folded and bedded into the mortar?

I don't know. It isn't apparent from the picture, I imagine?

The seam needs to be sealed also. A driving rain can enter that seam and find it's way inside. The top doesn't appear to be embedded into the mortar. If nothing is done about that, the top should be sealed with roofing cement.

That makes sense. But this guy wants to redo the whole thing. I haven't gotten an estimate from him just yet but he seems to be turning it into a big job.

It might not be a big job, but it needs be a thorough job, which you don't have.
I'm another advocate for step flashing with bedded tops.
This is not an area of compromise, and should be hose tested before signing off. Globs of tar are not a good sign-not even necessary if done well.

In what way does it not look thorough? I'm a novice at this.

"In what way does it not look thorough?"

It could be the picture, but I don't see flashing to bring water out of the shingles on the sides. The bend in front( over the shingles) might be inadequate in driving rain.
As maryland irisman points out, the seams on the side need sealing, preferably soldered.
Looking at a picture, one could be completely wrong in their assessment- but having an on-site roofer have doubts, plus your own, sorta clinches the issue.

I guess the question is this: if there is no step flashing, does that need to be installed? That is what he would want to do. I just spoke to him today and he said it would be $500. He said "no more than $500" but later he seemed to mean "at least" $500.

The way its laid under the shingles also could be the issue. Step flashing lets any water come out on the next shingle. It looks like they may have laid the flashing on the roof and then stepped the shingles on a flat surface. May not have much of a lip so water runs on the roof sheating. no way to tell unless its pulled up. I would have it redone, if the water leak is not stopped the repair later will be extensive and expensive.

http://www.thisoldhouse.com/toh/article/0,,194076,00.html
This link shows how a chimney should be flashed.

Well he came back, went up on the roof this time and sure enough, no step flashing and flashing not embedded in the mortar. Still no estimate, and he wants to come back again when it rains or with a hose, just to confirm.

I'm wondering now how much this is going to cost me, money being a bit tight around my house nowadays amazingly enough.