In this paper, we investigate how convective instabilities influence heat conduction in the intracluster medium (ICM) of cool-core galaxy clusters. The ICM is a high-beta, weakly collisional plasma in which the transport of momentum and heat is aligned with the magnetic field. The anisotropy of heat conduction, in particular, gives rise to instabilities that can access energy stored in a temperature gradient of either sign. We focus on the heat-flux-driven buoyancy instability (HBI), which feeds on the outwardly increasing temperature profile of cluster cool cores. Our aim is to elucidate how the global structure of a cluster impacts on the growth and morphology of the linear HBI modes when in the presence of Braginskii viscosity, and ultimately on the ability of the HBI to thermally insulate cores. We employ an idealized quasi-global model, the plane-parallel atmosphere, which captures the essential physics – e.g. the global radial profile of the cluster – while letting the problem remain analytically tractable. Our main result is that the dominant HBI modes are localized to the innermost (≲20 per cent) regions of cool cores. It is then probable that, in the non-linear regime, appreciable field-line insulation will be similarly localized. Thus, while radio-mode feedback appears necessary in the central few tens of kpc, heat conduction may be capable of offsetting radiative losses throughout most of a cool core over a significant fraction of the Hubble time. Finally, our linear solutions provide a convenient numerical test for the non-linear codes that simulate the saturation of such convective instabilities in the presence of anisotropic transport.