It might seem surprising, but as far as I know there is really not that much to say beyond: factor as fast as you can and check. (And this might be the 'truth' though I do not think there is a proof showing the equivalence of the two problems.)

And, this problem is also well studied as it arises in Computational Algebraic Number Theory;
for example in Zassenhaus Algorithm do compute integral basis for maximal orders to decompose a discriminant into squarefree and squre part is the main computational bottleneck.

I do not know what this order $n^{1/6}$ algorithm is, but this is (as you say) worse than factoring for large $n$.

Cf. http://mathworld.wolfram.com/Squarefree.html between equation (2) and (3) for a reference for about what I say.

Added: The reference provided by Igor Rivin is very interesting and pertinent to the question, to save rushed MO-readers some time, I extract some info from it.

The '$O(n^{1/6})$ algorithm' runs more or less along the lines sketched out already by OP.
Since checking whether an integer is a square is easy, it suffices to detect factors up to size $n^{1/3}$.

Now, this can be done Pollard--Strassen algorithm that allows to detect factors of size at most $B$ in time $O(n^{\varepsilon}B^{1/2})$, which for $B=n^{1/3}$ yields the claimed thing (up to the epsilon and this can improved a bit, but things stay slightly above $n^{1/6}$).

The running time for the squarefree-proving algorithm under GRH is of order $\exp( (\log n )^{c+o(1)} )$, where this $c$ is defined in a certain complicated way, and is conjectured to be $2/3$.

For comparison, and as mentioned in that reference, there are factoring algorithms with *expected* running time $\exp( (\log n )^{c'+o(1)} )$ with $c'$ equal to $1/2$ (Elliptic Cuves, Quadratic Sieve) and $1/3$ for Number Field Sieve.

So for this squarefree-proving algorithm one can *prove* (under GRH) something on its running time that one cannot prove for factoring algorithms.

Thus, it seems to me, if one is mainly interested in actually establishing/computing that a number is squarefree or not (as opposed to needing provable information/bounds on the difficulty of the task) it might still be better to go with factoring, as one expects it to be faster; and if ever the expectations should be wrong then, under typical circumstances, one could still do something else.