Google fully understands that users are reaching for incognito because they want their session to be 100% ephemeral they just don't care because they're paid to not care. Technical distinction between local and remote data is unrelated, Google could offer privacy if they wanted to.
Google has specifically created an entire page telling users exactly what to expect from incognito browsing: it's the first page that opens when you go incognito mode, every time
Do you think browsers should send to the server that it is in incognito mode? Because that is what you are asking for, that would just reduce privacy not increase it.
But sure, sending that extra tidbit of information specifically to servers that they can verify won't track you would be a good tradeoff.
Even worse is that they probably do try to detect if you're in an incognito session, but only for their benefit.
Edit: Here's an easy thing they could do. Even if we accept or pretend to desire cross-browser correlation at all, new-seeming browser profiles could have their information siloed for a few days and if they disappear in that time it all gets treated as an incognito session.
