Now, gossip can be recursive like AI. Does that mean that AI is about as reliable as gossip?
No; much more reliable, and one can further increase reliability through correct and careful usage.
The problem with AI isn’t AI being stupid but AI users using AI stupidly.

Scratching my head and thinking... hasn't religion already gone down this road? I.E created a set of doctrinal truths that serve as the measure of all subsequent truths. Just as the AI increasingly turns to itself to inform its answers, theism has already been doing that for thousands of years. In questioning religious ideology theism simply turns to religious ideology. What's the chance that AI will be smart enough to avoid a trap that humanity couldn't?
There are several flaws with your argument:
1. Several major religions are non-theist, for example, many varieties of Buddhism, and also the Jain religion.
2. There are numerous belief systems akin to theistic religions which likewise have doctrine, for example, Communism has various denominations such as Anarcho-Communism, Marxist-Leninist thought, Maoism, Hoxhaism, Stalinism and so forth, many of which have detailed complex eschatology and religious ritual which could be likened to religious praxis of a liturgical or theurgical nature. Some even have relics*
3. The use of AI is the use of software, and if you want to talk doctrinaire software use, look at fanatical Mac users or Linux enthusiasts (I myself love both platforms and there are Linux and Apple fans who hate me for it because I violate the ideological purity of their technological-aesthetic convictions). Thus far I haven’t seen anything comparable among AI users in terms of the kind of impassioned polemics of OS/platform userbases. And its not just operating systems - this paradigm extends into programming languages, execution environments like the JavaVM or Microsoft’s .NET platform, and even text editors, for example, the long running rivalry between emacs and vi, more specifically between GNU Emacs and vim, which also includes odd groups like those people who root for ed or acme or other eccentric editors.
*Albeit artificially preserved using chemical embalming vs. the naturally incorrupt relics of various saints in the Eastern and Oriental Orthodox and Roman Catholic forms of Christianity.