So the point of Lambda is that you don't have to run any infrastructure.

If Lambda had an open source equivalent, the "other pieces" are also needed to make it useful. The "Open Sourceness" is only one factor in that environment - it is at least as important that you can somehow call them from a stateless service and get adequate performance.

That tends to push towards all the services being run in the same environment (although it doesn't have to be like that - it's just easier)

Because you now either have to run those open-source alternatives yourself or be at the mercy of another vendor to provide those for your as a service, and personally my trust in pretty much any other vendor would be much lower than my trust in AWS by default.