I believe this is done on purpose as I had discovered the same issue with mine when I was on LP. A good friend of mine has a Weber Silver C Series with a side burner and it does the same thing. I think the intention was that if the burner was allowed to run too low, it might blow out. Since the grill comes as an "LP" model, the gas is heavier than air. If the burner was to blow out easily, the gas could build up somewhere lower than the burner and then it could migrate to the mains if they were on. The next thing that could happen is obvious. When I converted to natural gas, I found the low on my side burner to be much better. Of course, natural gas being lighter than air, if it blows out, the gas just dissipates upwards. I don't know if that is a lot better, but at least I know I won't get an explosion. A couple of things you could consider, one would be to convert the grill to natural gas. The other is to make the valve smaller. The way to do this is to first remove the front cover of the grill by removing all of the knobs, then the screws holding each valve to the cover. Next, remove the four screws holding the panel on (two on each end) then you can pull the front panel forward, exposing the wires attached to the igniters. Once you gently pull the wires from the igniter modules, the front cover is free to be removed. Now, you can disassemble the front of the valve for the side burner with two small screws then remove the stem and front of the valve. When you do this, be careful not to lose the spring or "D" ring when you remove the stem. Once this is out, you can remove the brass cone inside the valve, it simply pulls out. If you use needle nose pliers to do this and to hold the cone, this will keep the grease on the cone for smooth turning. If you look at the cone, you will notice a small hole and a large hole. The large one is for the high setting and when the valve is in this position it allows the flow of gas to be controlled by the orifice. When the valve is turned to the small hole, this hole size controls the low setting. I did this successfully on my friends Weber and it worked out nicely. Once you have located the small hole, you have to make it smaller. The only way I have seen this accomplished without replacing the cone, is to solder the small hole shut, and then re-drill it to a smaller size. You will need a numbered drill bit set for this project and (Of course) you will want to turn your gas off first and disconnect and remove the tank just to be safe. I have included a link below, which will show you pictures of what a similar gas valve looks like apart. It is from a Members Mark, but the internal portion of the valve is identical. I would recommend probably starting with a #70 and trying that first. If it blows out too easily then increase it (lower drill number) one size at a time. A #70 should yield about 5500 BTU's with 11" of LP. I am running a #60 on my low setting which yields about 5750 BTU's at 7" of NG, so this should be fairly accurate. I hope this helps and there is also a chart to show you these BTU figures at the link below. Good luck. Here is a link that might be useful: BTU charts and valve pics

Did you look for insects making a home in there?

Every so often, you read about people fully restoring a decades-old gas appliance. There apparently is a market for things such as old O'Keefe & Merritt stoves. And yes, from what I recall, this sometimes requires replacing the enamel coating. So, I know that this is possible. I don't have any idea whether it is cost effective and who would actually do this type of work. But if I was in this situation, I'd start by calling up places that restore old O'Keefe & Merritt stoves and ask nicely if they can offer advice. They might or might not be able to help you directly. But at least, they could probably point you in the right direction. When you have your answer, please report back here. This would be a useful resource to know about for other Houzz forum members.

The OP wrote: "- We can have a 10" duct, but can not install external blower because there will be at least 3 ninety degree turns for the duct to go through the roof (kitchen is on the 2nd floor and there is another floor sitting above the kitchen)." I must have missed this thread. Just what effect is being assumed here such that the location of the blower is affected by the three 90-degree bends, so long as none are right at the blower. (Hint: Pressure loss capabilities of kitchen hood system blowers are minute factions of an atmosphere. Due to this, pressure losses on the blower intake side and pressure losses on the blower output side have the same effect, particularly when a 10-inch duct is to be used for a sub 1000 CFM requirement.) If one wants [relative] quiet in an open area. the blower should be remote from the hood and a silencer placed in the intermediate ducting. As widely noted here over the past decade or so, rated CFM and actual CFM differ by a significant factor, and one cannot address kitchen ventilation with only hood system parameters. The make-up air system parameters are equally important.