UEFI is motherboard firmware, just as the traditional BIOS is.
I very much doubt that you will find new any of the cheaper laptops and pre-packaged desktops that come with Windows pre-installed without UEFI only booting as I seem to recall Microsoft demanding its OEMs make these available with Secure Boot enabled for Win10 pre-installs. Some of the more expensive such machines more intended for use by gamers and developers _may_ still support Legacy (or CSM) boot mode, but I expect that it will soon become impossible to avoid UEFI on new machines in this category unless manufacturers support generic Linux installs. The only exception I've heard of, though there may be others of course, is Purism which sells laptaps with coreboot firmware specifically for a Linux derivative (they package PureOS). Dell supports Ubuntu on some models and System76 sells machines with Ubuntu pre-installed, but both of these have EFI+Legacy firmware.
For motherboards and barebones systems sold without drives, as well as server hardware, the firmware seems to be more likely to support both UEFI and Legacy. I suspect that this will only continue until Microsoft decide that disabling Secure Boot makes a firmware off limits to Windows installs. Whether coreboot can be more widely installed I haven't yet investigated...
FWIW I have a current model Intel NUC (bottom end model but not the one with Win10 pre-installed!) which has firmware that supports Legacy fallback if the boot drive isn't setup for UEFI.
BTW, was there any firmware update ever released for your problematic Asus laptop (e.g. to add Win10 support)? If there was and you haven't done so already, it may be worth trying it.
I very much doubt that you will find new any of the cheaper laptops and pre-packaged desktops that come with Windows pre-installed without UEFI only booting as I seem to recall Microsoft demanding its OEMs make these available with Secure Boot enabled for Win10 pre-installs. Some of the more expensive such machines more intended for use by gamers and developers _may_ still support Legacy (or CSM) boot mode, but I expect that it will soon become impossible to avoid UEFI on new machines in this category unless manufacturers support generic Linux installs. The only exception I've heard of, though there may be others of course, is Purism which sells laptaps with coreboot firmware specifically for a Linux derivative (they package PureOS). Dell supports Ubuntu on some models and System76 sells machines with Ubuntu pre-installed, but both of these have EFI+Legacy firmware.
For motherboards and barebones systems sold without drives, as well as server hardware, the firmware seems to be more likely to support both UEFI and Legacy. I suspect that this will only continue until Microsoft decide that disabling Secure Boot makes a firmware off limits to Windows installs. Whether coreboot can be more widely installed I haven't yet investigated...
FWIW I have a current model Intel NUC (bottom end model but not the one with Win10 pre-installed!) which has firmware that supports Legacy fallback if the boot drive isn't setup for UEFI.
BTW, was there any firmware update ever released for your problematic Asus laptop (e.g. to add Win10 support)? If there was and you haven't done so already, it may be worth trying it.