Wifi smart devices can have that sort of completely local control, not just zigbee. LIFX devices use local control if you don’t connect them to the cloud. However, you’re then limited in lots of ways, such as you can’t then use a smart switch from a different manufacturer to control the lights. Home Assistant takes over the job of Google/Apple Home, which allows different manufacturer’s devices to all work together harmoniously. Those services also provide things like automations, turning the lights on when your smartphone arrives within a geofence for example. HA can do even better because those automations will work across Android as well as iOS. It also maintains the advantage of just one app to control your entire home.
As well, as far hardware, I think you’re misunderstanding a bit. Nabu Casa, the org that controls the open-source HA project, sells a couple of pre-built devices that run home assistant already. They’re designed as turn-key solutions for people with less technical know-how, and provide a bunch of expansibility so people don’t waste money needing to upgrade. The proceeds from those go back into supporting the projects costs. But you can go out and buy a Raspberry Pi Zero 2 W (512mb RAM) for $15 and it’ll happily run the kind of basic setup you’re after. But you will almost immediately run into it’s limitations if you try to do anything more complicated.
My HA server is running on an x86 VM with 2c/4t and 8gb of RAM to itself. Have a full music server running on it serving ~6 devices around the house though. Edit: 6 fixed devices. It can also be cast to a bunch of places from mobile devices. My music collection is in FLAC so it’s transcoding to lossy on the fly where needed.