i had one of the early 'netbooks', a 10in model. slightly shrunken keyboard to fit the form factor. it sucked for typing.
probably better off with a tablet and one of those cases with integrated stand and bluetooth keyboard. and you wouldn't need a bottom-of-the-barrel model, either, to keep with a $300 price point for everything.
Eh, I think it depends on personal preference. I had an ASUS eee pc, second gen device I think and if I remember correctly it was a 10 inch one. I had no issues typing on it. In fact I wrote my thesis on it, in LaTEX, 100 pages. No problem, I liked it.
I've never had a keyboard on a device under 13" that I could type comfortably on. You could have the best actuation and key travel in the world, but it doesn't make up for the keys being smaller than The minimum spacing on my fingers.
If that's all you're doing, just get a Chromebook. Has a history of success.
Those $200 machines are built to be reliable for kids to ruin them. I bought one five years ago and I'm just waiting for it to die, but it wont. It's slow for anything other than web browsing. But it was a great solution for movies and bypassing paying for hotel entertainment.
The obly smaller than notebook device that ever worked for me is the GPD Win Max 2023. The keyboard actually works and the performance is stellar. Of course the price is a while different matter but it doubles as a high performance gaming desktop especially with eGPU
Everything else so far looked great but was entirely impractical