I do trust the Linux distro maintainers that they don't have nefarious purposes. But they can't and won't verify third party projects' code, nor the huge number of contributors that come and go on any of these projects, or their transitive dependencies.
As has been shown, it's almost trivial to get malicious code merged into open source projects, so not really sure where your "trust" comes from. It's not trust, it's naiveness.
The proof is in the pudding at the end of the day, how many privacy scandals Debian had vs how many privacy scandals Android had? One model seems to clearly work better than the other. Talk is cheap, I like to see the results.
And to answer your question, of course they can't check everything, that's why it's a model based on trust and not a model based on verify.
What would happen if let's say VLC would upload your user documents in the background? They would get nuked out of the repository and never be seen again. That's why apps do not tend to do that.
I'm not against sandboxing and a strong technical model myself, it's just that if I have to pick between a trust model and technical features, well the trust model wins hands down 10 times out of 10 as it has a better proven track record.