How does one verify an app is running of its published code source?
How can users confidently verify that a FOSS application is running from its published source code? Is there a easy way to check this, or is this based of checksum and hashes?
There is, its called a Linux distribution repository or F-Droid for Android.
Why would you trust a badge on github if you don't trust the original author to not tamper with the binaries? The only way is to either built from source yourself or get the binaries from a trusted 3rd party that does that.
AFAIK most Linux distributions don't (yet) have fully reproducible builds (simply because it's a HUGE ask). For Debian for example, it's continued ongoing work: https://wiki.debian.org/ReproducibleBuilds
That is an unrelated topic. By using a distro you are already implicitly trusting them over the random author that publishes binaries on their repo or so.
Yes, I agree with the trust and that it's effectively equivalent for most users. But I read OPs reply as asking specifically for "a more automatic way of checking (that it was build from source cleanly)".
I think expecting that this has already been cleanly solved years ago is not unreasonable (until you realize how many rabbit holes this whole topic has to go down to work well).