For now, your side cart comes from Elementor, in order to disable it, navigate to Elementor -> Settings -> Integrations -> disable the “Mini Cart Template” option.
WooCommerce is responsible for updating the content in the mini cart. We have checked that issue on the default Storefront theme and there the issue persists. So you need to contact with the Elementor Pro support team for further solutions because without it the cart works well, you can be sure of that by disabling the Elementor Pro plugin.