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.
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.