Having a short run of fence incorporated across from the breezeway entrance would work. I wouldn't attach it to the house at all but set it forward to function as a privacy wall and sound barrier. It could have your house numbers incorporated into the face in a distinctive manner, and could also have power run to it for a lantern (good for security also) I can't say I'm a big fan of the yellow stone though, so the fence could be built from cedar as an alternate. I wouldn't recommend French doors or a glass wall unless you were completely upgrading your exterior.

Willits Compound · More Info

It does seem like a waste of space. But back to the posted photo - it simply depicts a nicely executed privacy wall - it has no relationship to your home specifically. Sorry for the confusion.

You might put up the wall (stone/brick/concrete will cut sound best I believe) & then have plants in front of the wall. Here're some pix that are close to what came to mind when I read your explanation, obviously plants need to be appropriate to your climate...

& in my image search, I came across this page of Houzz photos...

[https://www.houzz.com/photos/concrete-fence-phbr0lbl-bl~l_38922[(https://www.houzz.com/photos/concrete-fence-phbr0lbl-bl~l_38922)

If you build a gable over your existing garage, to match the
main portion of your house, you would add depth to the face of it. Therefore,
you could enclose with windows or French doors.

This is the exact method that was used in the house I mentioned in my previous post. It made a very nice and sun-filled room.

the idea of french doors on both sides so that people can still walk through to the backyard doesnt appeal. but you could close in some (2/3?) of the breezeway leaving just a narrower passage next to the garage and add the rest of the space to your house...mud room? laundry? without doors in front and out the back..which does scream 'closed in breezeway' to me.

and then dig up the cement forecourt and put in some nice landscaping.