The "green button" in Visual Studio is quite literally always related to debugging. If you do not want to debug your application, you build it and then run the application. I you do not want to debug your website, you use a webserver to serve it. Windows has IIS Express for that (and Visual Studio uses it in the background), but you can also use php's built in webserver if you want to quickly test.
I suppose it would be possible to ask it the dialog about installing xdebug should not be shown again, but it really doesn't make sense. Visual Studio is not a webserver, it's an integrated development environment.
Easy installation guide on IIS Express+PHP: https://articles.runtings.co.uk/2015/06/starting-local-instance-of-iis-express.html
PHP's integrated webserver: https://www.php.net/manual/en/features.commandline.webserver.php
If you want to be able to quickly start the web server from VS, you can either create a link to a script starting/stopping it under "View -> Other Windows", or use WhackWhackTerminal for Visual Studio.