The first website that comes up from your Google search has a pretty interesting take, "In the old days (B.C: before computers) when calculations were done by hand, analysts would use the normal distribution if the degrees of freedom were greater than 30 (for 30 df, the proper multiplier is 2.04; for 60 df, it's 2.00). Otherwise, the t distribution was used. This says as much about the availability of tables of the t distribution as anything else."
I find that to be a pretty plausible explanation for the $n \ge 30$ t distribution rule. One interesting property of the t dist with 30 df is that the most its pdf ever differs from that of the normal is .0045 units. If you want to match that closeness criterion with chi-squared approximation, you'd have to set $k=3425$. You could choose other closeness-criteria and get a much different answer. From a practical standpoint, it's worth noting that the chi-squared approximation maintains quite a bit of skewness - even for fairly large $k$. For instance, when $k=30$, the mode is $-.26$.