I'm a little surprised that you are finding websites that don't work if you disable third party cookies. Cookies are normally used to store information so it is still there when you change pages. This is often done with login details, for example so you can be identified and don't need to re-login on every page. It can also be used to store preferences, such as default values for search boxes etc. But all of this should be 1st party cookies coming directly from the site you are visiting.
3rd party cookies are typically issued from analytics and banner networks, so they can track you not just between pages, but between sites, and remember your likes/dislikes and deliver targeted ads etc.
A page shouldn't be storing anything it needs to function in a 3rd party cookie.