I created a Loom video to show the exact process (forgive the part toward the end where I fight with Loom’s controls over the setting I need). Here are the exact steps:
Spin up a fresh install of WordPress
Add and activate the Create Block Theme plugin
Create and activate a blank theme
Go into the editor
Select the header template part
Select the outermost group of the header template part
Toggle on “Inner blocks use content width”
The inner blocks do NOT use the content width that the plugin automatically sets in the theme.json file.
Consequently, if I add a NEW group block, then add blocks inside it and turn on or off that setting, it works just fine. It’s only the group block in the template part file that isn’t working.
By default, the outermost Row block in the Header template part is set to “Full width” alignment, which means that it will stretch to fill the width of the entire browser window.
If you’re not looking for it to be displayed at the full browser width, could you try selecting that Row block and choosing “Wide width.”
By doing that, the header width should then respect the “Wide” content setting that’s set site-wide in your Styles > Layout area.
Let me know if this accomplishes your goal or you’re looking for a different result. Thanks!