As I understand it, the mute tabs feature was primarily created to target advertisements. You're using one tab, and then an ad suddenly starts talking on a tab in the background. It's not a media controller, it's for media that CAN'T be controlled.
The current implementation is only problematic in my mind if you have so many tabs open that the button takes up the entire area. It only shows up when there's actually sound coming from the tab, and there's an animation to emphasize the clickable area.
One thought might be to move it to a right click context menu. make "mute tab" the first item on the list if there's sound coming from the tab.
The devs can't assume you have an adblocker (well maybe they can now, even if it's rudimentary), so until the built in version becomes good enough that it's never a problem, it has to be prominent, since there's a only fraction of a second to grab your attention as you reach for the tab.