Web browsers have to solve a very similar problem.
No, they do not.
When you put a game on a different platform, you rely almost entirely on your game's data to make that game work. Those fonts you see? Those are shipped with the game, either as bitmap images or as a font file that gets rasterized at load time. Generally speaking, games do not use system fonts. Or pretty much anything provided by the system; they do as much themselves as they possibly can.
Web browsers do not. They use fonts provided by the system. Oh, they may install their own, but users will expect them to be able to use system fonts.
Users also expect browsers to conform to the general UI requirements of the OS in which they operate. Close buttons should look like the close buttons on any other program of that OS. Dialogs should be laid out the way OS dialogs work in other applications for that OS. Same goes for scroll-bars and the like.
Most games make their own dialogs. They make their own UI elements like buttons and scroll bars. They look and feel as is appropriate for the game they're making, so that it aesthetically fits into the game's appearance.
At the end of the day, videogames are a matter of artistic expression. Web browsers are functional tools first and foremost. And thus, they need to behave like they belong on that OS. A one-size-fits-all approach just isn't a good idea.
Not to mention, the platform-specific aspects of a browser are probably the easiest part of them. So even if they took a one-size-fits-all approach, it wouldn't simplify development.
And wouldn't this also mean better rendering performance with animations utilising 3D acceleration, etc.
Hardware acceleration is often used in aspects of various web browsers' rendering engines.