Crank no start, not getting fuel to the fuel rail?

Hello, do you have the model with access to the fuel pump under the back seat? Did you check for power at the access cover electrical connector, or can you hear the pump running? It might just be another bad pump or sending unit, if you hear it running and theres no fuel pressure, it might be a stuck check valve on the sending unit, I will look up the pump and the part number to see if they are the same, Was the "used" pump taking out of a parts yard vehicle?

This vehicle has a Fuel Pump Ballast Resistor too, if it is below spec or has burned up it will not limit current flow at lower engine speeds and that may be why the first pump failed. Ill post the specs and location so you can check it. You can see it here in the 1st diagram. You will need a multimeter set on the Ohms setting to check the resistance of the fuel ballast resistor.
here is a picture of it 4th diagram with a built in heat sink, this one has a bracket on it. Its just a 2 wire resistor,, it looks to be in series with the fuel pump if the ecm opens the fuel pump relay contacts at low rpm to lower the current flow. Although the ballast resistor is only 0.3 to 0.35 Ohms, which isnt much. I would need to read up on this type of circuit. They dont build them like this anymore I dont think. The Fuel pump resistance is only 0.2 to 3.0 Ohms. So at 3 Ohms and 12volt supply, its only a 4amp draw.
The ES 300 fuel pump is a different part number (6th diagram) and does not use a Ballast Resistor in the fuel pump circuit. The ES is also a different engine.

https://www.2carpros.com/articles/how-to-check-fuel-system-pressure-and-regulator

https://www.2carpros.com/articles/how-to-use-a-voltmeter

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

Okay, to check fuel pressure on this vehicle you need to tap into the fuel inlet hose, or you can pull up the back seat where the fuel pump access hole is, take off the pump outlet hose there and run a hose out of the vehicle and turn the key to On, there should be a 2 second prime before starting the engine, this gets the fuel pressure ready for engine starting. If there is no fuel coming out, you'll have to check for power and ground to the fuel pump at the same location, (access hole). It is easiest to just unplug the pump and put a test light or multimeter across the 2 wires at the fuel pump connector. This will verify power and ground in 1 test. I'll post the wiring diagram and wire colors. But with the rear seat removed you should be able to hear the fuel pump prime with the key on or cranking. They are pretty loud. If you don't hear anything, just do the power and ground check. And I recommend checking the Ballast Resistor as well, if it is burned up, and you replace the pump, it will just burn out the new pump. I will post the service info in a couple minutes.

Okay, I see, have you done a pressure test yet? I know it's coming out the line, but if it's not making it up to the fuel rail, there has to be either a blockage in the line, the line is bent somewhere, or that pump is just not strong enough to push fuel up to the rail, possibly a check valve issue in the pump assembly. Another possibility is there is a corroded connection somewhere and the pump is not getting enough current. If it's only getting 2amps for example it might be able to pump out of the tank, but it's not enough to get full volume. Remember that one in there is a different part number. You can try pulling the line off the rail and off the pump and use a little compressed air to see if there's any issues with the line running up to the engine.

Do you have access to an amp clamp to see how much current that pump is pulling? I would also try unplugging the Ballast Resistor and see if that makes any change, the pump you have in now is not designed to run with that resistor in the circuit, service info stated the resistor was there for low engine rpm. I've been looking through service info for more on the operation of this setup. You say there is no 2 second prime for it, so we don't really know if the ECM uses that resistor during start up or not. I find it odd there is no prime of the fuel system, is that something you read or just noticed? If the fuel pump relay contacts open, current can still flow to the fuel pump it just has to through the ballast resistor first and then to the pump, it's a strange design, but if there is excessive resistance in the ballast resistor harness plug or the ballast itself it would slow the pump down dramatically.

The resistor is to limit current flow, not fuel pressure. Electrical current flow. Although it is a very low resistance, Ballast. Having it out of the circuit completely will rule out any circuit issues to or from the Ballast resistor. The ECM controls this circuit, so having it out of the way, you'll know it's something directly related to the pump, and yes, the pump in the ES 300 doesn't have a ballast resistor circuit. This is the FP circuit from the ES 300, it also has a regular Toyota engine in it, 1MZ-FE is a very standard engine in Toyota's. Whereas your Lexus IS300 has a 2JZ-GE engine. I would also check the resistance of the Ballast to see what it is. It's very low spec 0.30 - 0.35 Ohms at 68degrees F.
Another issue with the fuel pump can be a check valve in the pump assembly, or a cracked line in the tank, so it could basically be pumping back into the tank when it's under a load trying to pump fuel down the smaller lines to the engine. Thats why I will almost always put an amp clamp on the power or ground wire at the fuel pump and see how much the current draw is.

This is the fuel pump assembly for the ES, that metal section I think you're talking about is part of the pressure regulator. What you can try doing to see if the pump will hold pressure and not just drain backwards into the tank is take a piece of clear hose a few feet long and attach it to the pump output, power up the pump and allow the hose to fill up with gas and block off the end, see if it holds the fuel in the hose. If it just drains backwards back into the tank, then the check valve in the pump is stuck open or there's a crack in the assembly somewhere. The fuel pressure should hold for at least 5 minutes. Since you don't have a pressure gauge, I think you'll find the fuel is just running back and not able to build up pressure. Plus, you need to really get the correct pump in there. The ES pump should be putting out 45-50psi in just a few seconds.
An amp clamp is for measuring amperage(current) on the fuel pump or any circuit.
This is a current waveform from the Ignition coils from my truck (2nd diagram), this is from the main power feed to the coil pack. It's a V6, so each cylinder is firing. You can see an issue with the 1st coil firing, that negative amperage spike. This is a quick way to find failing or failed components.
The next one is a relative compression test on my Honda CRV. Same idea. If you look closely you can see 2 cylinders that are lower than the other 2. Bad head gasket on this one.