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The evolvement of wireless communication services concurs with significant growth in data traffic, thereby inflicting stringent requirements on terrestrial networks. This work invigorates a novel connectivity solution that integrates aerial and terrestrial communications with a cloud-enabled high-altitude platform station (HAPS) to promote an equitable connectivity landscape. Consider a cloud-enabled HAPS connected to both terrestrial base-stations and hot-air balloons via a data-sharing fronthauling strategy. The paper then assumes that both the terrestrial base-stations and the hot-air balloons are grouped into disjoint clusters to serve the aerial and terrestrial users in a coordinated fashion. The work then focuses on finding the user-to-transmitter scheduling and the associated beamforming policies in the downlink direction of cloud-enabled HAPS systems by maximizing two different objectives, namely, the sum-rate and sum-of-log of the long-term average rate, both subject to limited transmit power and finite fronthaul capacity. The paper proposes solving the two non-convex discrete and continuous optimization problems using numerical iterative optimization algorithms. The proposed algorithms rely on well-chosen convexification and approximation steps, namely, fractional programming and sparse beamforming via re-weighted $\ell_0$-norm approximation. The numerical results outline the yielded gain illustrated through equitable access service in crowded and unserved areas, and showcase the numerical benefits stemming from the proposed cloud-enabled HAPS coordination of hot-air balloons and terrestrial base-stations for democratizing connectivity and empowering the digital inclusion framework.