Your expression for the curl is wrong. You're working the curl out with the determinant mnemonic that only works for Cartesians. Analogous determinant expressions exist for other co-ordinate systems, but you need to look them up. You'll find that, in particular, the $z$-component is:
which vanishes aside from at the singular point at $r=0$.
The fundamental way to think about the curl is as the operator that, given specification of a plane, returns the circulation around a loop in that plane per unit area, in the limit as the loop's enclosed area tends to nought. That process gives you the extra $r$ factor that annuls the $1/r$ in your expression.