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.
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.