The intrinsic fine structure splittings (FSSs) of the exciton states of semiconductor quantum dots (QDs) are known to be the major obstacle for realizing the QD-based entangled photon pair emitters. In this paper, we present a theoretical and computational investigation of the excitonic fine structures of droplet-epitaxial (DE) GaAs/AlGaAs QDs under the electromechanical control of micromachined piezoelectric actuators. From the group theory analysis with numerical confirmation based on the developed exciton theory, we reveal the general principle for the optimal design of micromachined actuators whose application onto an elongated QD can certainly suppress its FSS. We show that the use of two independent tuning stresses is sufficient to achieve the FSS elimination but is not always necessary as widely deemed. The use of a single tuning stress to eliminate the FSS of an elongated QD is possible as long as the crystal structure of the actuator material is in coincidence with that of the QD. As a feasible example, we show that a single symmetric biaxial stress naturally generated from the (001) [Pb(Mg1/3Nb2/3)O3]0.72-[PbTiO3]0.28 (PMN-PT) actuator can be used as a single tuning knob to make the full FSSelimination for elongated DE GaAs QDs.