Hard starting when gas tank low

No stations will carry Car Talk where I live, in the North Georgia Mountains. I therefore have to listen to it on my computer. I have just finished listening to the most recent show during which a lady described how the engine in her Honda seems to be reluctant to start running when the fuel tank is low and more eager to start when the tank is full. This triggered a memory in what passes for my brain. In the seventies, my ex and I bought a 64 or 65 Pontiac from a friend who owned a garage. He told us that he had helped one of his ex-employee mechanics get the car. He said the car did not run when they got it. He allowed the mechanic to do a complete rebuild of the engine, a 400 cid, with a two barrel carburetor. The rebuild was done after hours, using parts the owner supplied. The mechanic turned out to be an alcoholic and was fired right after the engine was finished. When we got the car the engine was sort of reluctant to turn over when cranking. Once it started cranking, it fired right up and ran well. The garage owner told us that the engine was not broke in yet as all of the parts were brand new. He said it would crank better after it had been driven a while. That never happened. My wife was the primary driver of this car. One night, we were going out and decided to use her Pontiac. I got into the driver’s seat. I turned the key to start. The engine went Urrrr, urrr, urrr and in general gave me the impression the battery was on its last legs. I turned to her and said, “It’s got a dead battery.” She said no, just push the gas pedal to the floor while you crank it. I said, “No way is the gas pedal connected to the starter or charging system.” [I also probably thought silently about crazy women.] She repeated her stupid instruction. I ignored her and tried to crank it again. Same result. Then I thought what the heck, and pushed the pedal to the metal and hit the crank. The thing cranked like a brand new car and fired right up. It was the same every time one of us drove it as long as we had it. I have wondered for years why that worked.