A possible scenario: Your BIOS is set to PEG/PCI. The motherboard will first attempt the PCIe port. If the video card is not working properly at this point, then the motherboard automatically switches to the integrated graphics. Now the motherboard will only send video signals to the integrated graphics, and won't try the dedicated graphics card.
But maybe it's true that during initial POST, the 7750 is not getting enough power due to some limitation with the motherboard. Yet perhaps later on the PCIe port could receive the additional power that would enable the 7750 to work. But at this point the motherboard is no longer trying to use the 7750. So at this point, perhaps the 7750 is actually potentially working, yet the motherboard sends no video signal to it.
I'm just speculating. But if the above is true, and you could somehow force the signal to always go to the PCIe port, then perhaps it would work? Perhaps you can completely disable the integrated graphics, then maybe that would force the motherboard to send the signal to the PCIe port.
Just a speculation as I said. If you disable the integrated graphics, and then the 7750 still doesn't work, then you have no working video card. But you can probably find some way to reset it so the integrated graphics works again.
Since the video output will work when you switch cables to the integrated graphics, that means that the motherboard is sending the video signal to the integrated graphics. But the signal should be sent to the 7750. Perhaps the motherboard will automatically sense where the monitor is plugged in and send the video signal there? Otherwise the signal was being sent to the integrated graphics already, and then the 7750 could not possibly output a video signal since it is not receiving one.
According to your other post, the 7750's fan is working. So it kind of seems as if the 7750 is getting power, but the 7750 is not being detected as working. Or the motherboard doesn't even try to run the signal through the 7750. Or it tried, but failed, and then switched to the integrated graphics?
Since there is a PEG/PCI in BIOS, and the setting is set to the correct PEG/PCI setting. Then that would indicate that the motherboard should have no problems running a dedicated graphics card in the PCIe slot, no matter what PCIe generation that card is, as long as this card follows the rules/standards for PCIe regarding backwards compatibility.
If the article about PCIe 1 supplying power differently and less efficiently than PCIe 2+ is true, then maybe that could mean there is a potential benefit for a video card manufacturer to build its card aimed at working well with PCI 2+ while not working or having some problems when using some PCI 1 motherboards?
Or perhaps it could be the case, that earlier motherboards could not output video during post. Perhaps this could cause the motherboard to think the 7750 is not working, switching to the integrated graphics instead?
You're motherboard only has one PCIex16 slot right?
During POST, the internal speakers make sounds. This is a code to sound off if everything is ok or not. Look up what these sounds mean, and compare this to what you hear.