This problem is not found on all cars which have the pump in the tank, because all vehicles do not have the same fuel mixture. Depending on the quality of their break in, and the type of oil that they use; some engines run substantially richer than others. (I'm not talking about vehicles with compression, fuel or ignition problems here. These starting complaints come from vehicles which, when tested, meet all the manufacturer's specifications.) Only the leanest running vehicles (which incidentally are the ones that have the best compression) tend to be so lean during starting that there is not enough extra fuel available to start easily when the key is turned too quickly and the fuel pressure is below normal.
You might appreciate this more if you read about the fuel mixtures which were normally used in the 1950s and early 1960s, before emission control laws came into effect. In those days, many vehicles idled with air fuel ratios of 11:1 or 12:1. Those motors often put out 1,000 parts per million of unburned hydrocarbons; but started immediately when the key was turned, and idled as smooth as glass. When emission control laws were implemented, and air/fuel ratios were leaned out to 14:1 or higher, the unburned hydrocarbon levels dropped by 90% or more; but then people began complaining that their cars would not idle as smoothly as they used to, and that they would often stall after they were first started. So emission controlled fuel calibrations created a bunch of driveability problems. It took many subsequent years of experience, along with much experimentation and research on ignition systems, camshaft profiles, fuel delivery systems, and combustion chamber design, before those problems were largely corrected; but some of them have still never really been fixed. And during the 1990s, as in tank fuel pumps became standard, while emission laws became even stricter, and reformulated and oxygenated fuels became increasingly common; more and more people began experiencing starting problems.
Perhaps you can find a different way to fix those cars which now run properly when warm, and do not show any problems in diagnostic tests; but are hard starting at the times when the fuel pressure has not built up to normal. If you ever do; I'll be the first to congratulate you. But until that day; I'm going to continue to tell people who ask about the only answer that yet has been proven to work.