I've finally had a breakthrough tonight with my 5K monitor that has a DisplayPort 1.4 (DP1.4) port for receiving 5K @60Hz.
To recap: this is a non-brand Chinese made 5K monitor that has an LG 5K panel (same one that's used in LG Ultra Fine 5K monitors and in 5K iMacs). This monitor has the following ports:
USB3 (DP1.2 only - 4K @60Hz maximum)
DP1.4 (5K @60Hz)
HDMI 1 (4K @60Hz) maximum
HDMI 2 (4K @60Hz) maximum
In the fall of 2020, I purchased an M1 MacBook Air and quickly discovered that the M1 MacBook Air could not even detect that an external monitor was connected when I was using the same TB3 to DP1.4 cable and the same DP1.4 port in the same monitor. So, not only could I not send an image from the M1 MacBook Air to this monitor, but the M1 MacBook Air wouldn't even detect that an external monitor was connected (as it was evident from the macOS System Information utility > Graphics/Display, i.e only the built-in display was detected by the M1 MacBook Air.
So, tonight it occurred to me that there was one adapter (of many, trust me), in my desk drawer that I've never tried. That was a TB3 to dual-DP1.2 adapter, which is meant to either connect two 4K monitors via DP1.2 (one monitor into each port on the adapter), or to connect both ports to a 5K capable display via two DP1.2 ports on the monitor to stitch two signals into one 5K @60Hz image. For that to work, the 5K display must be capable of receiving two DP1.2 signals (each signal via its own DP1.2 port on the monitor) and then internally stitch that image to make it 5K. That's an old technology (before DP1.4 standard was released) that was the only way for a monitor to display 5K without having to use Thunderbolt. With the release of DP1.4 standard, it became possible for a monitor to receive a 5K @60Hz signal via a single DP1.4 port. My Chinese-made no-brand monitor has the ability to receive a 5K @60Hz signal via one DP1.4 port, but it has no ability to stitch a 5K @60Hz signal from two DP1.2 signals. That was the reason I never opened that TB3 to dual-DP1.2 adapter.
So, to my great surprise, when I inserted the TB3 to dual-DP1.2 adapter between the M1-based Mac and my monitor connected one of the adapter's DP1.2 ports to the monitor's DP1.4 port with a DP1.4 to DP1.4 cable, my monitor came on instantaneously, and not only did it come on (for the first time) via its DP connector when connected to the M1-based Mac, but the signal was 5K @60Hz. Now, the most I could hope for the signal to be 4K @60Hz because both ports on the TB3 to dual-DP1.2 adapter are both DP1.2 ports, which are not capable of more than 4K @60Hz. But, strangely enough, the signal received by the no-brand Chinese-made monitor via its DP1.4 port was a genuine 5K @60Hz. Trust me, by now I know how to tell for sure. I've verified that the signal was in fact 5K @60Hz (both in the monitor menu utility and via the macOS System Information.
Now, I thought maybe the TB3 to the dual-DP1.2 adapter was mislabeled, and perhaps the ports on this adapter are not DP1.2 ports but DP1.4 ports. That would explain how my monitor is able to receive the 5K @60Hz signal from the M1-based MacBook Pro via this adapter. So, I pulled out my 13" 2020 MacBook Pro (Intel-based 10th generation CPU) and connected it to the same monitor via the same TB3 to dual-DP1.2 adapter, and guess what I found out? You will not believe it. The 13" 2020 MacBook Pro (Intel-based 10th generation CPU) was sending a 4K @60Hz to the monitor, just as I would have expected it to because the TB3 to dual-DP1.2 port adapter is not supposed to be able to send a 5K @60Hz toward the monitor out of a single one of its two DP1.2 ports (because the DP1.2 specification maxes out at 4K @60Hz).
So, I don't know exactly why this adapter solved my issue with the DP1.4-based 5K monitor, but it works perfectly. Moreover, the monitor wakes up as soon as I press a key on the external keyboard. There's no unpluggin/pluggin back and dancing with a tambourine around it for 5 minutes required. It works absolutely perfectly with the M1-based MacBook Air. The magic is in this TB3 to dual-DP1.2 adapter. My suspicion (and it's my suspicion only) is that Apple wired something wrong from the GPU in the M1-based Mac to the USB4 port. I actually suspect this is a hardware issue and not a software issue. Somehow, this TB3 to dual-DP1.2 adapter not only mitigate this incorrect wiring in the USB4 connector on the M1-based Mac but it also allows for the 5K @60Hz signal to be passed to the external DP1.4 based monitor out of the adapter's single DP1.2 port (which is not supposed to happen, and which is not happening with the Intel-based Macbook Pro).
For the rest of us, here's the information about the adapter that fixed the issue of DP1.4-based monitors not working with M1-based Macs:
Vendor: OWC (Other World Computing; aka MacSales.com)
Part Number: OWCTB3ADP2DP
Name: OWC Display Adapter Dual DisplayPort with Thunderbolt 3