A gate will certainly enhance the privacy for the back yard fireplace. The arched material on the wall and the half pergola coming off the fireplace are both a bit odd.

It doesn't look like you have much room for planting a tall screen either. This is a situation where a roof is probably your best option for more privacy.

So get rid of the arched stuff on the wall, the half pergola on the fireplace and the tree by the wall will have to go. Now using the wall as one set of posts, build a new much larger pergola to cover the entire outdoor living space. The neighbor's wall side can have some screening between posts and the roof can be as open or as dense as you like. The roof could even have a movable fabric top. The trickiest design part is going to be how a new bigger pergola interacts with the taller chimney and the back of your house that appears to be at an offset angle.

This is a better shot to show the whole fireplace.

Not sure if it’s even possible. Or fits with your style. But I’m thinking a long planter with some greenery. Here are some pics I found rlly quickly

The half pergola on the fireplace needs a post on the right - that is truly odd, but it is a mighty fine start to a larger pergola to roof over the space.

Fireplace needs a hedge, screen or wall behind it so it doesn't look so exposed. And there isn't the distraction of the view beyond.

Planting in first example of a "long planter with some greenery" doesn't look like it has too many years before it has to be redone - roots in containers become congested, potting soil becomes worn out and broken down whenever plants are placed in containers. So if at all possible if you can plant a hedge there where needed in the ground instead you will be much better off.

I would not put a hedge behind the fireplace. That is THE VIEW you want to keep open when relaxing in this space.

Whenever something large and eye catching is placed in front of a view then the question becomes are we supposed to be looking at that something or the view behind it? It's just basic composition - you don't put competing subjects together in the same view.

I guess I am just the kind of person who prefers a long view with distraction over a monotone when I am outside. My neck still moves so I can focus on the firepit when I need to.

Great space! But my eye wants to edit the distractions to stone and mortar, plantings, and views. Could you do your grilling inside the fireplace and not have the shiny metal intrusion? Can you conceal the trash can? I don't see the problem with the tree. It looks like it provides shade and privacy from the neighbor's window. If you have lights on it (strings or an uplight), that will give you privacy even in winter as the neighbors will not see beyond the lights, at least in the evening. I'm confused by the arch you want to get rid of. Do you mean the arch of the hearth opening? How does that conflict with your style? What is your style?

ideas ( if you live in Zone 6B you may want to consider a dwarf Southern Magnolia):