Your problems are with Microsoft, not Intel. And, you should have known how to manage this limited amount of storage. Installing all of the browsers you have, plus who knows what other software, pups, etc, it is not a wonder that it is running "slow" now.
You recognised the problem two years ago, and decided to do nothing? Why? You should have immediately contacted Microsoft support to get assistance in cleaning up your drive.
It is not Intel's responsibility to support you on how to run Windows 10. Microsoft has a support forum for that: Results in Windows - Microsoft Community
Contact Microsoft, or the Linux community if that is really what you want. And, your "M$" references are old...
Just an opinion.
First of all, Microsoft releases updates on a regular basis. In the time between when a system is built (and loaded) and when it gets to an end-user, these updates can build up in this way. There's no one to blame; it is going to happen with every single system that you purchase with Windows already installed on it, regardless of the manufacturer in question.
Secondly, regarding space management, I have two comments. First of all, have you ever used the Windows Disk Cleanup applet? If not, learn man! Secondly, the NUC5CGYH has a drive bay into which you can add a 2.5" SATA hard drive for additional storage. There are drives available with up to 2TB of disk space that can be put into this drive bay. You can add a drive and, when installing tools and applications, specify that it be installed to this bigger drive instead of the (restricted size) system drive.
Next, if this update is the Fall Creator's Update you are talking about, you can specify that the update use space on a secondary drive during the installation. It may mean downloading the image from Microsoft and putting it onto a USB flash disk to start the process, but it can be done.
Finally, you are correct, you can install Windows from scratch onto an added hard drive and then configure the BIOS to boot from this hard drive instead of the eMMC. Simply specify to use the hard drive for the Windows install location and proceed. The installer will automatically pick up the Windows License key from the BIOS. Once this install is completed, you can go into BIOS Setup and change the Boot Order to boot from the hard drive first. Done. It's that simple.
My final comment? If you cannot handle something this simple, you haven't a chance of being successful using Linux.
Following Scott suggestion I'm also thinking that installing a secondary drive may provide some relief. I would add that it is possible to migrate the user's profiles that usually consumes a lot of space onto this secondary disk and keep the 32GB primary disk just for the operating system.
Eventually you may want to look into the possibility of using a tool such as Acronis to re-image your primary disk into a larger one.
even for only the operating system it is not that great.
i think it is possible, but you always have to keep an eye on it, that it still has enough free space.
32gb for windows is very very small.
i would put an bigger drive in it, and also put the system on it.
maybe even a small 256gb ssd or 512gb ssd, that would also be way faster than the small emmc.
i would not bother with this emmc to be honest.
at least if one uses windows i would not recommend using the emmc.
the best thing would be to clone it somehow on a big ssd and then forget about this problems.
i think acronis could do that, with the crucial ssd's there is even a free license that can be used.