Engine cranks but does not start

Do I understand the engine does run if you hold the accelerator pedal down a little? Was the alternator replaced in an attempt to solve this or did the problem start after?

I have a suspicion the battery was disconnected so the alternator could be replaced. If that is correct, idle speed is going to be too low. The engine may not start unless you hold the accelerator pedal down 1/4". You won't get the nice idle flare-up to 1500 rpm at start-up, and it will tend to stall at stop signs.

Fuel trim numbers and sensor characteristics will be relearned after reconnecting the battery as soon as you start driving, without you even noticing. The one exception is "minimum throttle". You have to meet a very specific set of conditions for that relearn to take place. Until that happens, idle speed will be too low.

To meet those conditions for the relearn to occur, drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals.

Also take a look at this article:

https://www.2carpros.com/articles/engine-stalls-idles-too-high-low-after-dead-or-replaced-battery