Solid-state sintering is a materials processing technique in which a compact of powder particles is heated to a temperature below its melting point, causing the particles to bond and densify through diffusion-driven mass transport. Most phase-field models for sintering build on the seminal work of Wang, where microstructure evolution is governed by coupled Cahn-Hilliard and Allen-Cahn equations. In order to account for the densifying shrinkage effect emerging from grain boundary diffusion, this framework introduces the concept of sintering forces. To deliver the particle rigid-body motions, these forces are converted into prescribed advection velocities for the phase fields. While computationally convenient, this approach bypasses fundamental continuum-mechanical principles and leads to non-physical effects, such as an artificial dependence of the shrinkage rate on the number of particles in a packing. In this contribution, we address these modeling limitations by proposing a continuum-mechanically consistent extension of Wang's phase-field sintering model. The phase-field evolution equations are fully coupled with the linear-elastic balance of momentum, where the sintering forces enter naturally as distributed body forces. This coupling yields physically consistent advection velocities for the phase fields and enables long-range mechanical interactions between particles to emerge in a natural way. The numerical implementation is realized within the hpsint framework, a matrix-free finite element code tailored for large-scale multi-particle phase-field simulations of sintering, featuring efficient block preconditioning strategies, implicit time integration, and advanced grain-tracking capabilities. Using a simple academic benchmark consisting of a chain of identical particles, we first demonstrate the deficiencies of the original Wang model and systematically analyze their origin. We then show how the proposed coupled phase-field--mechanics formulation resolves these issues. Finally, a set of two- and three-dimensional benchmark problems is presented to illustrate the improved physical fidelity, numerical robustness, and scalability of the approach, highlighting both the advances achieved and the remaining challenges in continuum-based phase-field modeling of sintering.