When trying to figure out how to do something the core does programmatically, a good place to start is the core itself. In particular, you want to look at Craft's controllers which handle all interactions with the UI and Craft's services which provide a lot of methods for common tasks.
In this case, you're looking for craft\controllers\SectionsController::...
"After pushing the exec widget all the codebase changes, fields, sections are moved to the Staging environment."
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Assuming you are properly using Project Config, you'll want to run craft up after deploying your files.
Your controller looks ok, though I would clean it up a bit and use more expressive variable names to reduce the surface area for bugs:
$templateVariables = [
'status' => true,
'message' => 'Sucess'
Now you will have access to the variables status and ...