I think you may be stuck doing exactly what you outlined, which is using an older system or building a system that will allow Windows 7 to be used. I've talked to a few people about this and looked at a variety of HP forum threads among others and I think the problem is not that the drivers don't work with Windows 10, because it seems like they do, but that the supporting software and utilities are based on code that was intended for use with Windows 7 and it seem to me like HP has simply abandoned the effort without bothering to update those applications and utilities with versions that are compatible with both Windows 10 and the printer itself.
Unfortunately without HP playing ball, it's almost impossible to resolve issues like this and since this is all Win7 era hardware and software, which means at best around 7 years ago, it's long past the point at which the manufacturer feels compelled to dedicate resources to resolving those issues.
Token efforts are all you are likely to see from them really. Sorry man, I don't have any other answers for you that are better.
The other problem is that with Windows 7, unlike Windows 8.1 and 10, the licenses weren't easily transferred to a new system if you installed a new motherboard because that effectively would have looked like a new computer to Windows. It can be done however. If you contact Microsoft, explain that you have had a motherboard failure and need to reactivate your Windows license on the new machine, it should not be a problem for you to simply connect your existing Windows 7 installation to the new hardware and have MS walk you through the activation process. Replacing a motherboard is not something that should permanently cause you to lose your legitimately activated copy of Windows so long as you still have that OS installation on disk or have your product key and can reinstall Windows 7.