It should still reach pressure then shut off. Depending on how bad the check valve is malfunctioning will determine how fast you lose pressure and then restart again, presuming no water is being used at a fixture.
A proper installation would never put the pressure switch at the bottom of a vertical run, even with an upward loop. The best installation is to put the switch on one leg of a tee with the center outlet of the tee going down a few inches to a cap. That leaves the sediment where it cannot cause any problems.
That should do it.
I've seen many 3/4 and 1 HP with remote capacitor as well as those same sizes with just a two wire input. I can't recall ever seeing a 120 volt submersible well pump, not saying they don't exist, but are not being used around here anyway. Would guess they would probably be limited to 1 HP or less.
Our water table in the valley I live in is at about 14ft (last I checked), we are over an ancient sand and gravel river bed with an adobe clay cap, damned near perfect for shallow well irrigation. 1/2HP 120V above ground jet pumps was about all anyone used around here for years, but 3 phase with little VFDs has now become more common. Still 120V input though because the HP requirement is so low. I just helped a neighbor with his, he got away with a 1/4HP pump and 10 gallon bladder tank to just feed his drip irrigation. Ironically I've never done it for myself, just too lazy I guess. I was going to have someone dig the well for me a couple of years ago, but when the drought hit, the demand went up so much that the local well drillers tripled their prices overnight (and are still booked months out). Now it's raining again, so I'll forget about it again, until probably April or May...
I've seen plenty of 120 volt jet pumps, just never a submersible pump of the type that typically gets dropped down a well casing.