Animals traverse vegetation by direct physical interaction using their entire body to push aside and slide along compliant obstacles. Current drones lack this interaction versatility that stems from synergies between body morphology and feedback control modulated by sensing. Taking inspiration from nature, we show that a task-oriented design allows a drone with a minimalistic controller to traverse obstacles with unknown elastic responses. A discoid sensorized shell allows to establish and sense contacts anywhere along the shell and facilitates sliding along obstacles. This simplifies the formalization of the control strategy, which does not require a model of the interaction with the environment, nor high-level switching conditions for alternating between pushing and sliding. We utilize an optimization-based controller that ensures safety constraints on the robot's state and dampens the oscillations of the environment during interaction, even if the elastic response is unknown and variable. Experimental evaluation, using a hinged surface with three different stiffness values ranging from 18 to 155.5 N mm rad, validates the proposed embodied aerial physical interaction strategy. By also showcasing the traversal of isolated branches, this work makes an initial contribution toward enabling drone flight across cluttered vegetation, with potential applications in environmental monitoring, precision agriculture, and search and rescue.