but literally nothing is stopping you from running your own server.
Nothing except gmail's very strict and hard to follow guidelines for spam filtering. Whether it's a byproduct of spam filtering or whether it's the intended result, the fact that Google essentially controls email traffic means you're not gonna have a good time communicating with others using your self-hosted email. This issue has been raised by self-hosters getting blacklisted, all the way to companies getting rate limited. If your intended use is to communicate with your everyday person, and considering the everyday person probably uses Gmail, you're in for a bad time at some point down the line.
They already have a sort of functional "Assistant". You can type to it, or you can use OpenAI's Whisper speech-to-text language model loaded up on your Home Assistant machine through an add-on in order to talk to it, it works pretty much as well as any other proprietary speech to text model, except it's self hosted. The assistant can talk back with another add-on, though the voices are still too robotic IMO.
Key part in all of this is the "sort of functional" bit. Commands seem to have to be very literal to be understood, otherwise it just tells you it doesn't understand.
I'd still rather host my own assistant than rely on Google or Alexa, though, so I'm just gonna put my faith on the HASS team.