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The thermal atomic vibrations of amorphous solids can be distinguished by whether they propagate as elastic waves or do not propagate due to lack of atomic periodicity. In a-Si, prior works concluded that nonpropagating waves are the dominant contributors to heat transport, with propagating waves being restricted to frequencies less than a few THz and scattered by anharmonicity. Here, we present a lattice and molecular dynamics analysis of vibrations in a-Si that supports a qualitatively different picture in which propagating elastic waves dominate the thermal conduction and are scattered by local fluctuations of elastic modulus rather than anharmonicity. We explicitly demonstrate the propagating nature of waves up to around 10 THz, and further show that pseudoperiodic structures with homogeneous elastic properties exhibit a marked temperature dependence characteristic of anharmonic interactions. Our work suggests that most heat is carried by propagating elastic waves in a-Si and demonstrates that manipulating local elastic modulus variations is a promising route to realize amorphous materials with extreme thermal properties.