That's really pushing it for a pump, especially a self-primer. I suspect that this is not actually self-priming, but manual priming. When you lift water to a certain height under a vacuum it can start to vaporize at room temperature, depending on the design of the pump, friction losses in the suction lines, temperature of the water, etc. 14 or 15' is a practical maximum, and for that everything has to be nearly perfect.
You said that the water you pour into the pump appears to be moving downstream into the sprinklers. There should be a check valve on the discharge of the pump to prevent this. If not, the pump won't be fully filled with water when you turn it on, and it can't generate enough lift to start the water moving.
There should also be an isolation valve downstream of the check valve so you can keep the pump full while it starts. If not install one. If installed then close it, fill the pump, start it then gradually open the valve.
I suspect what may have happened is that the water table in the area has gradually dropped, causing the old pump to lose its prime then it burned itself out.
Or the foot valve may be not opening all the way. This would cause additional friction losses in the suction line, and the water vaporizes before it can get to the pump.
Did you replace the pump with the same make, model and power rating? a more powerful pump will be more likely to give you a problem.
A better installation would be a submersible pump. These are multi-stage turbine pumps, shaped like a long cylinder about 3-1/2" or 5-1/2" in diameter with the motor on top, pump on the bottom and the water passes around the motor to a vertical discharge pipe. The hole thing hangs off the discharge pipe and drops down into either a 4 or 6" well casing, and sits a few feet off the bottom of the well completely submerged in the water. This does away with a foot valve and priming problems. The electrical feed is usually strapped to the discharge pipe, and you have a "pit-less adapter" that holds the pipe and keeps the pump in the proper vertical position.
Here's what they look like: