There are a number of real threats to Gmail. This isn't one of them.
That said, I'd bet that this threatens encrypted mail more than webmail. Have consumers ever chosen security over convenience?
Burned once, consumers will choose security over convenience. They'll even go so far as to choose inconvenience on the assumption that it makes them safer. Witness people's acceptance of inconvenient airport security, even as baggage went on totally unscreened.
The very core of what makes Gmail so enjoyable is that their expertise in free form search allows me to pretty much stop caring about organizing my mail at all. I generally just hit archive, and rely on search to retrieve whatever I may need to later. No web mail client can provide such a service against encrypted mail.
At the same time, it is Google's access to my mails' content that allows them to show me 'relevant' ads. Without this ability, providing Gmail for free will get a lot harder to justify.