How to upload local changes to a live server without overriding new posts/pages?

Making content (database) changes on a staging environment is never recommended. You should only use a local/staging environment for template changes. Then, update your theme/plugins (the wp-content folder), and push those files to the live site. Once that’s completed, you can then populate the content.

Database merging/synchronization is it’s own specialty/portion of the industry, and it’s very strenuous and fickle. It’s a workflow change, for sure, but you need to really consider the files and the database two separate entities in all cases. The piece you mentioning of “The client has added new content” is the exact use case that will happen every time, and there is no easy way around it except to avoid the conflict in the first place.