From the viewpoint of special Lagrangian geometry, the limitations of the Lotay-Pacini picture are:

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  The Lagrangians are largely relegated to the back stage. As clear from the above, in the Calabi-Yau case the space of critical points is too enormous. Their framework may be more useful in the negative Kähler-Einstein case, but the lack of ellipticity is a severe obstacle.

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  There is no attempt to link up with Floer theory. As such, they lack satisfactory existence criterions for the holomorphic curves, despite making some limited progress in special cases. In fact, Lotay and Pacini’s geodesics seem too oversimplified from the Floer theoretic viewpoint: one needs to address transversality questions in general, and holomorphic disc breaking should occur in moduli spaces of dimension $\geq 1$.