If the lower check valve does not hold, it will create a (partial) vacuum when the column of water falls. This vacuum is what will suck in air if there is a a suction-only leak or a hole in the pipe. Plastic fittings are prone to suctions leaks when the vacuum causes atmospheric pressure exerted against the outside of it to deform. Metal fittings don't generally deform enough to create a suction-only leak. A pitless adapter O-ring could possibly fail to seal under suction.
Even if there is no suction leak, with some water, the vacuum can liberate gasses that then can cause sputtering but the volume of gasses liberated is usually less than what you describe which leads me to believe there is a true leak.
Clear evidence of a leak is that the pressure will drop with no water use as the water runs back towards the well.