I didn't restate your equation I posted the CORRECT equation as it's used. There is no CFM in the equation as you added to yours.
I don't see how it's different from what I posted, but we can disagree.
The eq that your trying to adapt (p1v1=p2v2) to this story, is a SPECIAL case of the ideal gas law that is RESTRICTED to Static systems. The systems must be at equilibrium for it to work. In a dynamic pumping system you will have many other factors to deal with. Just for Fun why not go back to the general gas law eq. PV=nRT and deal with the delta T of 320deg F added to the gas as it's compressed from STP to 175 psi like my compressor does.
The Timgr eq as we can call it, as your its inventor

has yet to be proven by its creator. We know that it fails to explain the pumping seen by a real compressor. Because the CFM produced @ 180 psi compared to the value @ 90 psi is not 50%
Your response bothers me because I feel you are making this
WAY more complicated than it needs to be. Any physical problem can be drilled down to a point where the complexity becomes overwhelming - that's not profound. Making appropriate simplifications can be hard though.
Maybe we're simply getting off track. I thought we were trying to get some understanding of what the specs mean, not of what goes on inside the compressor. In terms of the specs, everything is spec'd in terms of volume at STP (mass) and the time dependence is not complicated. It's just a rate.
I presume you're pointing to the STP comment above, and you are agreeing with me, or I'm agreeing with you, or some such.
This one remains rather fuzzy what is the set point pressure?
It's the pressure that the compressor maintains in the tank, +/- the operational range; ie the setting of the pressurestat.
No you said that P*V=W Once again you have a STATIC system there! NO work done! There is no change in either pressure or volume ( no displacement) so no work done on the system.
I see that you have corrected your self below and called it correctly as POTENTIAL energy
Sorry, I fail to see the distinction. Energy is energy. The unit of work is energy. The equation is not invalidated because you can make the conditions more complicated. It is indeed true that the amount of work you can do with the air is equal to the stored work, which is the pressure times the volume. I can't change that. PE and KE are something of a distinction without a difference, though useful for a lay explanation (sorry if that sounds haughty).
We could instead do this in terms of power - we'll have to pick a unit, say K for kilowatts... then PV/t = K ... but that has a time dependence, so I'm in jeopardy of stirring up a whole new pot of ugly complexity.

(nothing snarky intended - I just wanted to use that icon!)
Let's do it anyway, just for fun. We know that conservation of mass says that we can't release any more gas than we can pump into the tank at STP. This is a suitable use for P1V1/t=P2V2/t by your rules, since it's a static situation. So P2 = 185 psig and V2 = 15/(185+15) * 19 = 1.4 CF. We can deliver power as 1.4 CFM of air at 185 psig. The easiest way to proceed here is to convert this to Pascals and Kg to get watts.
185 psi * 6.895 kPa/psi = 1276 kPa
1.4 CF / 35.5 CF/M^3 = 0.04 M^3
1276 kPa * 0.04 M^3 / min / 60 sec/min = 0.85 kW of power = K.
Now, 1 kW = 1.34 HP, so the compressor can deliver 1.14 HP as air to the tool at a constant rate.
This also gives us an efficiency rating. If the motor really produces 5 HP, then we can deliver 1.34/5*100% = 27% of the motor power to the tool. This is reassuring, since I would have guessed that the efficiency would be something less than 50%, but still a significant fraction of the motor power.
Cheers - Tim