Just adding to Valvecrazy's comments.
Self prime is a misleading term.
Only vacuum assisted "self-priming" pumps will self prime, all other so called self-priming pumps need to be primed initially and will re-prime each time the pump is restarted, provided the system remains primed.
It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. (Sherlock Holmes - A Scandal in Bohemia.)
If you look at the drawing in an earlier post you will se that normally a volume of water would be left in the pump house. So in that case i think that it would ne enough to get the ejector effect going had "suck" more water to the pump. Remember, this is a pump for domestic application so frequent start/stop but with relatively small volume.
On a domestic application a foot valve is mandatory and keeps the pump from losing prime. Even if/when a jet pump will self prime, it would take a minute or so. With frequent starts/stops of a domestic application, you don't want to be waiting for water as the pump re-primes after each start. Also, there are ways to keep a jet pump from cycling on and off so much on domestic applications.