Your toilet problem could me caused by many things. The first thing you should do is remove the lid and flush the toilet. Watch the water in the bowl to see if it rises when flushed. If the water level rises high in the bowl, then it is a drain problem. If the water level does not rise high when flushed, but just swirls, that means the water is not getting from the tank to the bowl fast enough. Your plumber could be correct, but you may need an acid that is stronger than vinegar to remove the built up calcium deposits. We use a product called Sizzle, which is hydrochloric acid. I don't recommend doing the treatment yourself. We do the treatments for customers, because it can be dangerous.

Test the toilets yourself. Rapidly pour a bucket of water in to the toilet bowl. If it flushes well, the problem is not in your sewer line nor in the lower part of the toilet. If it does not flush well, then toilet trapway or the sewer system after the toilet has a problem. If it flushes well with the bucket, then the water in the tank is too low or is moving out of the tank too slowly. Simple as that.

Take the toilets completely out. Set them on some grass you don't care if it dies. Get two gallons of muratic acid. Use with caution, this stuff will hurt you. Fill your bowl up to the water line. Soak for two days. Empty and rinse out, put your hand down the throat, and feel for mineral deposits. Repeat process if you feel any. You may also need to put muratic acid down the troat itself so it will penetrate the water outlets below the bowl lip. This worked for me when I rebuilt two wall hung toilets. Again use extreme caution when doing this, gloves, glasses, and work upwind of the muratic acid.
Your older toilets are worth saving compared to the new ones.

Thank you to all of you who offered advice! I think the problem is finally resolved. A friend who heads the maintenance team at a local hotel came over, adjusted the amount of water in the toilet tanks so it was higher, and presto change-o! The toilets now flush fast and completely. He explained that having more tank water creates a faster flush (don't ask me how, but I'm sure you guys would know!). Too bad I didn't ask him first - I could have saved hundreds of dollars in plumber bills. And I agree with you, gwilson2 - I don't want to give up my big ol' toilet!

Definitely too late but if you are paying a plumber to come, I think replacing the toilet is the best option. Once you pull the toilet, you have spent more money than a new toilet would cost. In many areas of the country, the payback on a new toilet would be pretty fast - because of 1.6 gallons instead of 4 or 5.

Now to mention the greater height and longer bowl which are worth the price of a new toilet to most people. I am really surprised that anyone would consider trying to fix a 1966 toilet....

Thanks for your post, David - I was surprised, because you are the first person who has ever suggested to me that the new toilets are superior to the old ones. Most people I know dislike the newer ones with lower water capacity, and wish the larger models were still around. Also they tend to sit lower, which is awkward for some people. I've noticed more recent models sit a bit higher now, though. I concede that they are more economical and environmentally friendly. But for now? I still love that old one, and it had worked perfectly for over 14 years until the recent problems. And since I'm a family of one, I don't feel too guilty about the water usage!