I don't know, this seems to be a really low-effort blog post. The given example is obviously contrived from the unreasonable improper (-\infty,\infty) prior and the low \sigma^2=1 likelihood. If it was really "pure noise" then you'd have \sigma^2=\infty which rightly gives you a flat posterior.

For sure Bayesian gives you more flexibility with your assumptions, so it's easier to shoot yourself in the foot. But when used correctly it can be more powerful, and often easier to interpret.