So if you disregard what the pattern looks like at the faucet, is the issue how long it takes to fill a cup with water?
You have two things, pressure and volume. If your pump is only 1 gpm that may be the problem. You could find out what the pump gpm is then compare it to how much is coming out the faucet.
If you use a stop watch and time your faucet flow and know your pump gpm, that will tell you if you have a pump problem or a restriction problem. You could have a pump that produces 500 psi but if the volume of that pump is only 1 quart per hour you would open the faucet and see an initial surge of water and it would quickly be reduced to a trickle. If you resized the faucet opening to say a hole only 1/64 th inch you would probably see a much improved solid jet of water.
Another example would be a diesel fuel injection system that uses 25,000 psi. If the injector with it's extremely fine holes was removed from the fuel line you would see a mere dripping of fuel. It's volume is extremely small but it's pressure is incredibly high.
Picture a finger pressing on a wall. Even when pressed hard (force) it will not leave a mark. Now picture a thumb tack with a small but flat end, it can be pushed into the wall with the same force. Picture two cylinders with pistons in them and equal forces applied to them, through the stroke of both of them one will move a larger volume of water but when the openings are closed the other will show a higher pressure on the gauge. Pressure equals force x area.