As announced in the press, Chrome v74 finally introduced a way to bypass websites which detect InCognito mode. Funny enough, none of the news sites actually bothered to check if it really does what it's supposed to do.
However, I've tried it (by enabling chrome://flags/#enable-filesystem-in-incognito) in multiple sites and they still detected the mode.
Even this short JavaScript detected it, and it even specifically checks for filesystem, not anything else.
Update: As of Chrome v75 (and simply by default since Chrome v76!), the only site I found that still detects this mode is Netflix. Does it mean sites like Netflix rely on other methods to detect InCognito mode? I read elsewhere it might simply be about running a DRM protected player that technically can't run in InCognito mode.