Just to first clarify, flashing only occurs when the Jetson is in recovery mode, and the flash is via USB. In that mode the Jetson is a custom USB device. During that flash, when actual flash completes, the Jetson will automatically reboot. This is when JetPack/SDK Manager will install optional softwave, e.g., CUDA, in which case the Jetson is no longer in recovery mode; instead, the Jetson is now fully booted. One trick is that the first boot login setup must be complete so that there is an admin account to run this installation through.
That optional software can be over any network connection. The default is to use the virtual wired Ethernet over USB. The actual wired Ethernet works if you tell JetPack/SDKM what IP address to use. Wi-Fi would also work, but you’d have to set the Wi-Fi up first for that admin user, and you’d have to tell JetPack/SDKM what IP address it is. Most of the time Wi-Fi also implies that user must be actively logged in to the GUI.
Incidentally, JetPack/SDKM allows you to uncheck “flash”, and simply install optional extras. You won’t have to reflash each time unless you are dealing with a true flash and not just optional software.
Can you ping the Wi-Fi IP address from the host PC? Can you use ssh to log in to your admin user account over Wi-Fi from the desktop PC host?