I did a restore on the exact same model. You could be getting into a money pit. To check the burners, you could disconnect the NG hose and hook up a LP regulator/hose to it and fire it up. All parts for this are going to be more expensive than parts for a Genny, but then again, that Summit was over $2K originally.
The biggest concern in my book is rust, both the cart and the firebox. Unlike the single cast firebox of the Genny's, the Summit's lower firebox is 4 pieces. The left and right are cast aluminum, but the front and back are porcelain coated stamped steel. The front part of the box tends to corrode where the burner tubes pass through. Remove the grates and the flavorizer bars and inspect the inside front of the firebox. Next, remove the burner control knobs and remove the control panel cover (the black sheet of metal with all the writing on it in your third pic). This will expose the gas valves and the ends of the burner tubes. Inspect where the tubes pass through the front of the firebox.
Here is what the inside of mine looked like
I still haven't gotten around to trying to treat the metal and prevent it from rusting anymore. But, I figured that it took 20 years to get to this stage, so it should have quite a bit of life left in it.
I saw in your thread at TVWBB that LM has a side table for you, I'd take him up on it. While most other parts are still available for this Summit, the U-frame support for the side table is impossible to find. Weber doesn't have any nor do any of the other replacement parts sites that most of us use.
The question of whether to restore or not. First question, do you want to run NG ? I mean, if you dump this one and go out and buy another grill, will it be NG or LP ? If you are set on NG no matter what, then that $1000 or whatever doesn't factor into the decision on this grill as you will be spending that no matter what.
If the firebox is in good enough condition, and the frame is solid, then it's up to you. A restore will probably run you $400+(+) which is more than it would cost to find and restore a nice 3 burner Genny, but then again, you'll end up with a nice 6 burner Summit.
BTW, back to the question of NG or LP. There are mixed opinions on this, but you can convert that Summit from NG to LP by replacing the orifices and adding a LP regulator and hose. If interested in going that route, talk to LMichaels about it in the other thread. Short answer is that it will work, but grill will run a bit hotter at low and medium settings than it would if you were able to replace the entire manifold (valves and orifices). To me, this hasn't been a problem as I rarely use low or medium, and the 6 burner Summit has a lot of real estate. If I need to cook something on low, I simply shut down some of the burners and go with indirect heat.