Browser independence

It worries me that I am now having to use Google's browser (Chrome) to access Google's products (such as gMail) because it is faster, and Microsoft's browser (Internet Explorer) to access Microsoft's products (such as Microsoft Office Outlook Web Access and Sharepoint) as Chrome won't even let me log into Microsft Office Outlook Web Access, and does not deliver all the functionality of Sharepoint.

Whatever happened to browser and operating system independence on the web?   I wonder if browsers will be more compatible across different operating systems than they are with each other?