Asus may be picky about recognizing USBs. Have the USB present when you shutdown, then in BIOS/UEFI Settings, it should show up in the bootorder, and may be put first. This setting may not persist even if you just switch USBs.
Did you hashcheck the downloaded ISO? A bad download may prevent the USB from booting. Does the USB work in other ports, in other machines? Maybe there was a bad write to the USB, and it wont work anywhere. Number of partitions on the hard disk may prevent an install, if DOS partitioning is used and all four primary partitions are used, but shouldn't affect booting the USB.
Setting up a bootable USB which works for both BIOS and UEFI is dog-meat simple -- a byte-for-byte copy of the ISO to the USB is all it takes (the dd method). You wont get any persistence, and any space on the USB over 4G will be wasted, so various programs offer ways to do that, confusing the basic issue. Additionally, about 5 years ago, the ISO added a link, which is fine for an ISO9660, but not allowed on a Fat filesystem, so just copying the files out of the ISO onto a FAT USB no longer worked (for a UEFI boot). Since the USB you made works on other computers, you have the technique, now you just have to figure out what your Asus needs to boot, or find a Windows dd equivalent to just copy the ISO, which boots both ways. The Rufus defaults should work, but I am not familiar with Rufus. I do know you should avoid offers of strange filesystems like ntfs or exfat23.