Is there any real downside to having syntax errors?
Obviously, that's not a problem if you already know exactly how different browsers will treat your code, or you're using parsing errors so elemental that they must be patched up identically for the page to work. For example, on the Google homepage, they don't escape ampersands that appear in URLs (like href="http://example.com/?foo=bar&baz=qux — the & should be &). That's a syntax error, but one that maybe 80% of the web commits, so any browser that couldn't handle it wouldn't be very useful.
Anyhow, one downside to having syntax errors might be that parsers which aren't as clever as those in web browsers, and which haven't caught up with the HTML5 parser standard, might choke on your page. This means that crawlers and other software that might try to extract semantic information (like microformat/microdata parsers) might not be able to parse your page. Google probably doesn't need to worry about this too much; there's no real benefit they get from having anyone crawl or extract information from their home page, and there is significant benefit from reducing the number of bytes as much as possible while still remaining compatible with all common web browsers.
I really wish that HTML5 would stop calling many of these problems "errors." They are really more like warnings in any other compiler. There is well-defined, sensible behavior for them specified in the standard. There is no real guesswork being made on the part of the parser, in which the user's intentions are unclear and the parser just needs to make an arbitrary choice and keep going (except for the unclosed center tag, because unclosed tags for anything but the few valid ones can indicate that someone made a mistake in authoring). Many of the "errors" are stylistic warnings, saying that you should use CSS instead of the older presentational attributes, but all of the presentational attributes are still defined and still will be indefinitely, as no one can remove support for them without breaking the web.
There is no reason to allow most of these errors other than coding sloppiness.