Balancing throughput and fairness promises to be a key enabler for achieving large-scale digital inclusion in future vertical heterogeneous networks (VHetNets). In an attempt to address the global digital divide problem, this paper explores a multi-high-altitude platform system (HAPS)-ground integrated network, in which multiple HAPSs collaborate with ground base stations (BSs) to enhance the users' quality of service on the ground to achieve the highly sought-after digital equity. To this end, this paper considers maximizing both the network-wide weighted sum rate function and the worst-case signal-to-interference-plus-noise ratio (SINR) function subject to the same system level constraints. More specifically, the paper tackles the two different optimization problems so as to balance throughput and fairness, by accounting for the individual HAPS payload connectivity constraints, HAPS and BS distinct power limitations, and per-user rate requirements. This paper solves the considered problems using techniques from optimization theory by adopting a generalized assignment problem (GAP)-based methodology to determine the user association variables, jointly with successive convex approximation (SCA)-based iterative algorithms for optimizing the corresponding beamforming vectors. One of the main advantages of the proposed algorithms is their amenability for distributed implementation across the multiple HAPSs and BSs. The simulation results particularly validate the performance of the presented algorithms, demonstrating the capability of multi-HAPS networks to boost-up the overall network digital inclusion toward democratizing future digital services.