Don't know why anyone would use SQLite in production except for as some KV store embedded in an app.
The vibe I get from this thread is that a lot of people have only experienced SQLite from the vantage point of something like Rails running in a "test" environment, and are used to explaining to people on some PHPbb somewhere that you can't really use SQLite to back a multiuser SAAS app. That's as may be, but when you're building infra components, you control who the "users" are; not everything is a web server taking requests from all comers.
Do you understand what kind of application Tailscale is? For all intents and purposes it's exactly what you described above.
Using openssl with a wrapper service? Doesn’t seem particularly dubious to me.
Coincidentally, one of tailscale's founders already built this: https://github.com/sshuttle/sshuttle