Would you enclose this?

The one in the OP's house just needs a good strudy fireguard attached to the wall behind the stove. The one she showed, was a pinterest picture and is only an example. I think you need to layout your room so that the furniture stops children running round it and playing games.

I grew up around stoves and open fires, some in tiny kitchens as they heated the water and, while I know there were accidents, especially with open fires. The only accident I can remember was when I was 13. when wool/nylon socks came in. On a cold day, I held my feet close to the stove to warm them and all the nylon in my socks melted. My mother was not amused. They were new socks.

I would leave it open as it will give out more heat. We have a wood burner that is in the fireplace which we have enlarge so there is plenty of room around the stove for the heat to circulate and not all go into the bricks. We have bought a fan which helps this and we also have a gadget which tells us if the wood is dry enough and safe enough to burn”

Thanks, you have all convinced me to leave it exposed.
We will invest in a fireguard for when the DC’s visit if we use it.

To be honest, I don’t think we will use it that much. The property has a modern heating system.
It’s a recent old workshop conversion and has building regulation certificates etc and a good EPC rating so was probably installed as a feature rather than a means of heating. I’ll let you know in the winter!

Barmyoldbat the heat that goes into the brick, comes out again and warms the house when the fire isn't burning. The bricks in the chimney act as a giant storage radiator.

Yes I know, our bedroom is able the sitting room and the wall is always warm but with some space around the fire you will get the best of both worlds, heat coming straight where you are sitting and the warmth from the bricks upstairs.