One natural attempt to compactify Teichmuller space is by the visual sphere of the Teichuller metric. However, Anna Lenzhen showed that there are Teichmuller geodesics which do not limit to $PMF$ (in fact, I think it was known before by Kerckhoff that the visual compactification is not Thurston's compactification).

However, it was shown by Cormac Walsh that if one takes Thurston's Lipschitz (asymmetric) metric on Teichmuller space, and take the horofunction compactification of this metric, one gets Thurston's compactification of Teichmuller space. In fact, he shows in Corollary 1.1 that every geodesic in the Lipschitz metric converges in the forward direction to a point in Thurston's boundary. I think this gives a new proof that Thurston's compactification gives a ball.

As Misha points out, it's not clear that the horofunction compactification is a ball.

Another approach was given by Mike Wolf, who gave a compactification in terms of harmonic maps, and showed that this is equivalent to Thurston's compactification (Theorem 4.1 of the paper). Wolf shows that given a Riemann surface $\sigma \in \mathcal{T}_g$, there is a unique harmonic map to any other Riemann surface $\rho \in \mathcal{T}_g$ which has an associated quadratic differential $\Phi(\sigma,\rho) dz^2 \in QD(\sigma)$ ($QD(\sigma)$ is naturally a linear space homeomorphic to $\mathbb{R}^{6g-6}$). Wolf shows that this is a continuous bijection between $\mathcal{T}_g$ and $QD(\sigma)$, and shows that the compactification of $QD(\sigma)$ by rays is homeomorphic to Thurston's compactification $\overline{\mathcal{T}_g}$ in Theorem 4.1. I skimmed through the proof, and as far as I can tell the proof of the homeomorphism does not appeal to the fact that Thurston's compactification is a ball, so I think this might give another proof that it is a ball.

One natural attempt to compactify Teichmuller space is by the visual sphere of the Teichuller metric. However, Anna Lenzhen showed that there are Teichmuller geodesics which do not limit to $PMF$ (in fact, I think it was known before by Kerckhoff that the visual compactification is not Thurston's compactification).

However, it was shown by Cormac Walsh that if one takes Thurston's Lipschitz (asymmetric) metric on Teichmuller space, and take the horofunction compactification of this metric, one gets Thurston's compactification of Teichmuller space. In fact, he shows in Corollary 1.1 that every geodesic in the Lipschitz metric converges in the forward direction to a point in Thurston's boundary. I think this gives a new proof that Thurston's compactification gives a ball.