Source code is necessary for trust, but not sufficient.
Having the source code available says a few nice things:
1. This company is confident enough to show their work
2. This company is "good" at software engineering (or it could reveal the opposite)
[1] I know some people can and do run their own mail servers. I can respect that, but I trust the Google devs and organization to be properly competent and incentivized to do a good job keeping my email account safe.