A couple of comments--you said
"But as soon as your pump exceeds the GPM of your source you will lose
your inlet pressure. The source will become atmospheric or less."

If you are using your house supply than as I mentioned above--you have an
infinite supply of (city) water. You only lose inlet pressure as a function
of the inlet line loss. Assuming that the house inlet pressure is somewhere
in the order of 50 psi than the inlet line loss has to be greater than 50
before it will start to have any impact on pump performance. Most positive
displacement pumps can operate without any flow loss with inlet pressures
as low as 5-6 psi below atmosphere. So, again, as noted above: "as long as
you have a positive inlet supply pressure to the pump it will put
out the specified flow." As a side note: cavitation is caused when
entrained air comes out of solution resulting in a mixture of air and liquid
being drawn into the pump inlet. This happens when the pressure (due to
line loss etc.) falls below the vapor pressure of the fluid. Have you ever
heard of V/L ratio (vapor to liquid)? Depending on the application, V/L
capability is typically included as part of a pump requirement.
With respect to operating on the pump relief valve. True, the pump is still
pumping it's spec flow--but where is it going?? NO Where! Just about all the
pump flow is doing nothing but going around in a circle from output to input
via the relief valve and all it's accomplishing is to generate a lot of
heat---what little isn't going through the relief valve is getting out to
wash whatever you're washing. Since- flow in equals flow out- inlet line
loss and inlet pressure is of little consequence. If you are operating on a
relief valve then there is a significant restriction downstream of the pump
discharge.
MLD

If you put a pressure gauge one foot away from the end of a
water supply that is plugged you will see 50 psi. Gradually open
a gate valve to create flow and what was once 50psi will start to
drop immediately. Once the flow is completely unrestricted you will
see 0psi on the same gauge. At that time you will have your
maximum free flow rate.

If you put a pump inlet at that same water supply, and that pump can
develop a vacuum, you should be able to draw slightly more flow, but
your gauge will be below atmospheric if this occurs.

The pump manufacturer suggest to have a free flow supply of 5gpm
for their 3.9GPM pump. This means they don't want negative pressures
at their pumps inlet. Cavitation can result. Some pumps have ceramic
plungers that wear great but are easy targets for cavitation damage.

As you have stated when the relief valve is activated the actual
flow at the inlet is reduced so that a less than adequate supply may
suddenly become adequate, however is your relief valve always
activated? It is not, and hence the manufacturers recommendations.