the term ‘Eucharist’ does not just refer to the consecrated bread, but also refers to the consecrated wine.
sharing the Chalice wasn’t unheard of, but it was very uncommon. the perception that only the priest drank the Blood prior to the changes stemming from vatican ii is not correct. however, the practice of only giving the Host became common over time. various writers (church fathers, saints, etc.) have attested to communion under both species in the past. at the council of trent, this very question was addressed. the council clarified the fact that Christ is present body, blood, soul, and divinity in both (each) species. it doesn’t matter, as far as the truth of the sacrament goes, whether you receive one or both. this is one of the reasons that only giving the Host was allowed to become the norm.
the second vatican council decided to return to the earlier practice. even though the sacrament is the same under only one kind, there is also sacramental value in consuming Jesus under both species. as you note, it is more scripturally significant to consume both. receiving under both kinds emphasizes the sacrificial nature of the sacrament, the reuniting of the body and blood in the resurrected Christ, and its aspect as the communal meal which creates and sustains the mystical body of Christ. one of the reasons for changing the posture for communion was to further facilitate the sharing of the cup.