> This is one thing I'm surprised the open source community hasn't developed a popular alternative for: Teamviewer.

That's because it's not only a matter of software but also a matter of service (STUN is not enough, TURN is required in many situations. And someone has to pay for the servers and bandwidth...). If IPv6 had succeded instead of IPv4+NAT, maybe the situation would be different.

In terms of software, the open-source x2go (based on nomachine NX libraries and protocol, see https://wiki.x2go.org/doku.php) has been very efficient in my cases when bandwidth was limited and when I had no big NAT issues.