ICloud limits your browser preference. This is because not all browsers work well with it. An app called iWork does give you access to certain browsers. These include the conventional safari, Firefox, chrome and internet explorer. However, iCloud may not full support the browser features. Hence, not all features of the browsers are expected to work with iCloud.
This is an indent problem in the system, and something that apple has implemented to limit third party apps from accessing the system and overtaking it. However, it has led to problems for users which is why apple is facing some backlash. The problem remains unsorted as apple has chosen to stick with its policies.