In this work, we present a computational study of the small strain mechanics of freestanding ultrathin carbon nanotube (CNT) films under in-plane loading. The numerical modeling of the mechanics of representatively large specimens with realistic micro- and nanostructure is presented. Our simulations utilize the scalable implementation of the mesoscopic distinct element method of the waLBerla multi-physics framework. Within our modeling approach, CNTs are represented as chains of interacting rigid segments. Neighboring segments in the chain are connected with elastic bonds, resolving tension, bending, shear, and torsional deformations. These bonds represent a covalent bonding within the CNT surface and utilize enhanced vector model (EVM) formalism. Segments of the neighboring CNTs interact with realistic coarse-grained anisotropic van der Waals potential, enabling a relative slip of CNTs in contact. The advanced simulation technique allowed us to gain useful insights on the behavior of CNT materials. It was established that the energy dissipation during CNT sliding leads to extended load transfer that conditions size-independent, material-like mechanical response of the weakly bonded assemblies of CNTs.