"Rather than using fallbacks and shims to ensure a design and layout look the same across all browsers, we’d provide the mobile vertical single-column layout to all browsers and then serve up advanced functionality to those browsers and viewport widths that can take advantage of them."
+1 This is a great solution that doesn't take a huge amount of time to implement. Esp if you only use CSS Grid for 2d layouts and use flex for which has decent IE11 support for laying out components (where mobile is often just stacked 100% width components).
A good way to bring a client round is to explain that the time saved on making IE11 pixel perfect is better spent adding features elsewhere.
We support Grid by default.
If Autoprefixer see that your CSS is too complicated for IE, it will warn you to disable Grid support and use some JS polyfill.
I've seen fallback CSS and JavaScript polyfills used.