You have already bought and paid for the N3350 machine so all ideas of what you should buy instead are pointless.
It is more powerful than a Raspberry Pi but the N3350 has only two cores. Forget virtualization and Proxmox.
For several reasons.
Your little new machine has the perfect capacity to run the HAOS with Home Assistant and a couple of Addons of the type that do not load the CPU too much. You will have a good reliable system.
Remember that HAOS is an operating system. Inside HAOS a number of Docker containers are running that makes up the working Home Assistant environment. HAOS is created for making a good solid bridge between the hardware interfaces like USB and Home Assistant itself.
When you run in a virtual machine then you add additional layers. Proxmox is built on Debian. And then you have the virtual machine layer. And inside that an additional operating system HAOS (based on Alpine Linux). Each layer is something that can break.
I have lost count of the number of reports we see every month where people upgrade something and say “my Conbee II stopped working”, “my Zwave2mqtt” lost all devices", “my Zigbee2mqtt lost all devices”. And notice - many run … Tata … Proxmox. And then downgrade and reload backups and loose even more.
Don’t. yes it can work. I have tried to run HA on Proxmox myself on a beefy 8 core i7 machine for development and testing of betas. And then it is not critical that it breaks
Home Assistant is a box that will run your home. When you have automated everything and turned your house into Starship Enterprise your entire life is depending on Home Assistant being reliable. And the most reliable way to run Home Assistant is with HAOS on bare metal, on hardware the HA developers use. Ie. a short list of Arm single board computers or a standard 64 bit 386 machine. Anything else is asking for trouble.
Personally I have ended up running my Home Assistant as HAOS on a Home Assistant Blue (Odroid) which is one of the short list of ARM single board. Rock solid! Only Addons are file editors and ssh.
And I have two more small computers.
An i5 based NUC running Debian 12 on which I run Mosquitto, Deconz, Zigbee2MQTT, Zwave-UI, Mailserver, and Unifi. N. ESPHome, Zigbee2PQTT and Zwave run in Docker. Rest on bare betal installed with apt. I run two parallel Zigbee networks but will probably phase out Deconz and put all on Zigbee2MQTT which continues to impress me.
An n100 based Asus mini computer running my Web site (lavrsen.dk) and my Frigate. Frigate needs lots of resources! if I did not have 6 cameras on Frigate I could have run it all on one computer. The new N100 is so nice. Lots of power and no fan!