Usually, one builds a patio near the back of the house entrance in order for it to be CONVENIENT. If there is outdoor cooking going on, one usually needs to have regular access to the indoor kitchen as part of the cooking process. If it's too far away, it is inconvenient. Also, if one just wants to sit on the patio and chat, one does not want to be inconvenienced by having it be too far away. Evaluate if the current patio works for you for these activities. I suspect it would not in which case you should plant to construct a patio where it makes sense. The present patio may be fine as it is for a fire pit getaway. You could leave it as is. Or you could improve/rebuild it if the rustic character doesn't suit your tastes.

Pour a concrete floor over that. Or build a low deck floor(one step up).

Gazebo. Or pergola. Hundreds of designs, octagon, square, round, how ever you wish. You can screen it for comfort, run an underground line for fan/lights

some ideas:

It’s not very big, but I think it’s charming. Enlarge it. Is there a backdoor from the house over near where we see the hose hanging? Enlarge the patio over in that direction. Make a sidewalk from your backdoor leading to it. Can you find more old bricks to match?

I don’t mind the grass growing up thru it, but there’s several ways to kill the grass.