I also use the NUC to do other things, so I have to use a virtualisation programme such as Hyper or Virtualbox, and in this case I am trying to figure out which one is best.
If Home Assistant satisfies me I might consider buying a dedicated device, I’ve seen that on the website they suggest buying their ‘Home Assistant Green’ device or a Raspberry
So you would install Proxmox on the NUC instead of Windows?
And then on promox you install a virtual machine with Home Assistant?
Yes. I absolutely would do this, given the options.
And you very likely could do another Windows virtual machine inside the same NUC box, under Proxmox, if you absolutely need Windows.
It is what many use and myself included.
Proxmox is a paid product, but their test service is free and the license allow it for this usage.
You need to make a small change that will cause a popup box to appear on connection to the proxmox administration service about not running with a production setup.
It is a click or two and it is gone and the proxmox administration service is not a daily thing to access, more like a monthly.
I was looking at the prices, it is a disproportionate annual cost.
At this point it is better to invest 100 euro and buy ‘Home Assistant Green’.
I’m not quite understanding what problem can occur in having “home assistant” on a virtual machine.
In the sense that I can use it for Home Assistant for free?
The other use I need the Nuc for is to make FTP backups to an external hard drive connected to the NUC, at the moment I am using FreeFileSync
Free alternatives to Proxmox could be XenServer (XCP-ng), VMware ESXi, oVirt … do you know?
Yes. It is free to use for Home Assistant as long as you are not running in a commercial setting.
Great then. No I am using it in my house where I have many different devices between philips, miboxe, legrand etc etc…
Would you be so kind as to tell me which link I should use to download Proxmox and the instructions for installing it with home assistant?
I am a Windows user, I need a guide to follow on linux 
There are multiple guides, both in text and video, for this.
Just make sure it is for Proxmox 8 or newer, which should be any guide posted in the last 6-8 months.
This could be one.
I prefer one without the scripts, so I know what is being done, but this one is easier.
I can not vouch for the scripts though, since I did not run those, but maybe others can.
I think that even if I’m given command by command, I still wouldn’t understand what I’m doing, just like with a script
What do you think of the official 100 euro Home Assistant product?
Its a decent option for running Home Assistant and some lighter addons.
Frigate and Voice Assistant will be a challenge though.
For the NUC drivers, how does it work? Are they installed automatically?
So I could for example also transform a Microsoft Surface with Proxmox?
Or if I wanted to buy a small device ONLY for Home Assistant, something cheap $150 for example… what could I get?
As long as the Surface has an UEFI BIOS, then yes.
You would need to disable SecureBoot, but that should be pretty doable.
On a Surface or a small computer I would probably not go for Proxmox, but rather install directly on bare metal.
The question is when a computer is powerful enough to warrant a Proxmox installation so you can use the excess power on other VMs/containers. It is question with many opinions and you should probably ask again here, when you have chosen a specific hardware setup.
I didn’t know that you could install Home Assistant directly on hardware like a NUC or a Surface, etc., etc…
What I was asking is, if I need to buy hardware specifically for Home Assistant, which costs around 150/200 dollars, without repurposing something I already own, what could I get?
I can not answer that, because I am not located in a place that use dollars. 
You probably have to start a new thread about that question and remember to state your location then.
This thread is marked as solved and with another topic, so many might skip the thread.
Neither do I, use euro for me would be under 200 euro
It was just to make the point, a device that costs about half as much as a Nuc
Raspberry I have seen that doesn’t cost cheap
Refurbished computers are sometimes used.
They are often cheap and powerful.
Laptops can also be a good platform, as long as you remember to check if there a setting to start the computer after a power failure. Some have it, but not all.
A side benefit with laptops is that the battery works as a UPS for the machine and it is often easy to get the power data into HA to make automation on.
I installed proxmox, it was very easy with the guide.
Now I’m trying to get the Wi-Fi card working without success… with chatgpt I’ve tried various commands and installing various components but nothing… the card is available but remains in the ‘Down’ state
wlp0s20f3: <NO-CARRIER,BROADCAST,MULTICAST,UP> mtu 1500 qdisc noqueue state DOWN group default qlen 1000
link/ether 50:76:af:c0:62:3c brd ff:ff:ff:ff:ff:ff
do you know a guide also for configuring wifi, which is activated when the lan network fails, and how to configure bluetooth as well?