If you're questioning the files (though there are many others that support even what is seen in the code above and there are database settings to boot), then upon submission of the form, the general process would include:
includes/auto_loaders/config.base.php which loads
includes/init_includes/init_cart_handler.php which loads
includes/main_cart_actions.php which is expected to load
All of the includes/extra_cart_actions files which may prevent "update" or change of the total amount/quantity before feeding back to have the shopping_cart page load again. After those files (if present) are parsed/acted upon then there is the shopping_cart class itself found in includes/classes. Then through all this there is the possibility of one or more observers that could be taking action to prevent changing the total quantity in the cart. There also could be a cart override "class" that is loaded ahead of the default shopping_cart class to take its place and that may have something involved.

Lots of possibilities. Would suggest identifying what has been installed that might have modified any of those files or that process, differences in the overall system, and anything else that might be further helpful.