Hi Barry,
The reason for the excessive noise is because of where Phoenix installed the tank in relation to the pump. They officially did nothing wrong. It's only that you gained half of the benefit with their installation. They likely installed the tank so far from the pump because there was no room to mount the pump and tank together. The tank needs to be very close to the pump to eliminate most of the pump noise. Using a flexible hose between pump and tank eliminates the vibration that apparently in your case is getting amplified through the hard plastic PEX plumbing between the two. The flexible hose is the same kind you would use to hook up a toilet or faucet at home. I prefer the stainless steel braided type.
My installation shown is odd but it works. It is extremely difficult to see, but I have a flexible stainless steel braided line between the pump and tank. The vibration created by the pump is cancelled away through the flexible line and also from the heavy tank. The hard plastic PEX plumbing after the tank feeding the entire motor home, does not get vibrated.

Concerning pump-run-time. With any accumulator tank, when you run the water and the pump kicks on, the pump is pretty much keeping up with the demand. If you turn off the water at the time the pump kicks on, the pump will run roughly 20 seconds. Keep in-mind that I never timed how long it takes for my pump to fill up the accumulator tank if it has been 100% deleted.
When the accumulator tank is 100% empty (at zero plumbing pressure) there has to be a certain PSI inside the tank. If my memory serves me well, it is between 17-20 PSI. It should be set to the same pressure that the pump turns on at, whatever that is. It could be that Phoenix did not set the tank pressure properly, or the air valve is not tight and it lost pressure. The valve is the same as a car tire, located on the opposite side of the pipe fitting, under a plastic cap. In my case, under the blue plastic cap on top.
I should re-take the pictures with my Samsung phone. That picture dates back to 2008, taken with real old digital technology.