It sounds a lot like vapor lock. From the picture you can see that long expanse of stainless steel fuel line going into the carburetor. I would start by replacing that with a non-metal hose or try wrapping the metal parts of the fuel system with insulating tape.
Is the fuel tank and line leading to the pump clean? Debris can temporarily restrict fuel flow under suction. A fuel line restricted with tar or rust can lead to vapor lock.
I'm not sure if that has a filter on the pickup tube on the float but I have other cars that the filter on the tube would collapse and the car would stall like it was running out of gas. Then would run fine at times.
I’m pretty familiar with early 50’s Chevrolet 6 cylinders and agree with those suspecting vapor lock…especially if the problem is made more frequent in hotter weather. But I do disagree with some of the suggestions…
I just don’t like the idea of sacrificing a section of metal line in that position above the engine to rubber. And IMHO an electric fuel pump increases leak issues and fire risks in the event of a leak . I’d keep both the stock pump and also keep the metal line as-is and insulate. Then I’d look to reduce radiant heat to it. You can do that with a shield but that would look odd. I’d rather have the exhaust manifold cerma-coated/heat coated. Your manifold looks nice but suspect it’s just header paint. Heat-coating looks very similar but does what the name implies. Then I would absolutely source only non-ethanol fuel. That alone will probable do more than anything else. And ethanol is an oxidizer and tends not to play well with some of the older rubber compounds, gaskets and seals anyway. Also, in my experience while not as common as when the car is stationary and/or just idling, vapor-lock CAN and WILL occur while driving.
A simple way to confirm the issue next time it happens…and to get it home, is to carry a bottle of water and an old rag. Soak the rag in the cool water and lay or wrap it around the section of fuel line just upstream of the carburetor. Wait a couple minutes and then try to start the engine. If it starts, it’s almost certainly a vapor-lock issue.
The only other possibility that comes to mind for your issue would be an intermittent sticking timing advance maybe???
I saw that too. A wild guess is that it might be a coolant manifold of sorts intended to warm up the carb before the days of an auto-choke? Looks like maybe the ports have been capped. Just a guess, but it looks like some kind of sender also goes into it. Curious, b/c under my 52 carb there’s a bake-a-lite insulator…just the opposite purpose. I’ve never personally seen anything like that on post-war cars.
Edit: I happened to visit my 93 year old dad this evening and showed him a close up but he didn’t remember seeing anything like that either. Maybe the OP or someone else can shed some light?
Back with my books. Make the setup as I described here, but instead of pumping for a minute, cranks until you get 1 pint of gas in the jar and time that. You should get 1 pint in 60 seconds or less. You should also measure the pressure by hooking a gauge to the line. Normal pressure is 3-5 psi.