The 31-year-old Sergio Perez showed flashes of brilliance at Red Bull in the 2021 F1 season, but Mercedes still won the constructor's title.
Lando Norris signed a contract extension with his team in May, with Sainz replaced by Daniel Ricciardo at McLaren in 2021.
In 2021, Williams had its best season since 2017, finishing eighth in the Constructors' Championship with 23 points.
Despite the team's competitive position, Haas F1 CEO Steiner underlined the necessity of keeping the squad motivated.
Sainz and new colleague Charles Leclerc rapidly built a strong bond, and the two teamed to place Ferrari third in the constructors' standings.
The Enstone-based team has spent many years in Formula One's middle, but their goal remains to return to the top of the field.
Russell has joined the 2021 Constructors' Champions to replace Valtteri Bottas, who has moved to Alfa Romeo for the 2022 season.
Despite Alonso relishing the new challenge, he disputes the notion that he returned to the grid in 2021 a happier and more relaxed driver.
The Aston Martin F1 team will start the 2022 Formula 1 season without team principal Otmar Szafnauer, who has departed from the role.
Williams boss Jost Capito believes George Russell's demanding nature will help him succeed at Mercedes alongside Lewis Hamilton.