Are you saying that sites that don't provide a positive experience across screen sizes shouldn't worry? Because browsers aren't supposed to do anything besides present information?
Sites don't worry, people do and plenty of sites are divorced from the people that created them. In the interest of keeping that information accessible the onus is first and foremost on the makers of the consumption devices and associated browsers to render that content as accurately as possible. Regressions (where content that rendered just fine on previous generation devices but where future devices for whatever reason fail to display this content) should be avoided where possible.
Note that in the case of content such as 'flash', 'silverlight' and other plug-in related content this is going to be a major problem.
Finally, in the interest of commerce and user experience website owners are free to improve their sites to enhance the experience for their users but they should not be required to get basic functionality working.
Note that in the case of content such as 'flash', 'silverlight' and other plug-in related content this is going to be a major problem.
Finally, in the interest of commerce and user experience website owners are free to improve their sites to enhance the experience for their users but they should not be required to get basic functionality working.