there was a startup that addresses exactly this (findo.io) but it is dead now - I guess they didn't get enough traction.

The idea to have 'personal google' is good, however it fails as cloud/proprietary solution. If I give access to my gmail/drives/messangers I want to be sure that this data will not be tracked/used by 3rd parties (even as 'telemetry') - this is possible only with open source self-hosted software.