Cates has also been a member of the United States men's national ice hockey team, serving as an alternate captain at the 2022 Winter Olympics in Beijing.

[5] Rather than entering college immediately after high school in 2017, Cates spent time in the United States Hockey League (USHL) to improve physically and develop his frame.

[11] That January, Cates moved to the top line with Peter Krieger and Nick Swaney, the trio combining for 10 goals and 18 points in their first 18 games of the calendar year.

[12] Cates's first multi-goal game came on March 1, when he scored twice and assisted once in a 4–2 win over the Miami RedHawks, earning National Collegiate Hockey Conference (NCHC) Offensive Player of the Week honors for his performance.

[15][16] The Bulldogs shut out the UMass Minutemen to win the 2019 NCAA Division I men's ice hockey tournament, with Cates scoring in the 3–0 championship game.

[27] Taking on the role of top-line center for the Bulldogs after Justin Richards left to play for the New York Rangers, Cates saw a dip in offensive production during his junior season, with five goals and 19 points in 28 games.

[29] The Bulldogs lost to the Minutemen in the 2021 NCAA Division I men's ice hockey tournament semifinals, with Cates recording an assist in the 3–2 overtime loss.

[30] Rather than leaving college to begin his professional ice hockey career, Cates returned to the Bulldogs for the 2021–22 season,[28] where he repeated as captain and as a Preseason All-Conference selection.

[43] In 16 games to end the season, Cates recorded five goals and nine points,[44] and he earned praise from interim coach Mike Yeo for his maturity and "really good hockey sense".

[46] Injuries to prominent players like Couturier, Cam Atkinson, and James van Riemsdyk forced Cates into a significant role as a rookie, where he served as a defensive-oriented two-way forward.

[59] He finished the year with six goals and 18 points in 59 games and told reporters that his season had suffered due to "an identity crisis and then some confidence stuff, some injuries", and citing his focus on offense instead of defense as a struggle.

[60] After serving as a healthy scratch for four of the Flyers' first five games of the 2024–25 season, Cates took over as a penalty kill specialist, drawing positive comparisons to Couturier for his defensive prowess.

[69] Cates's older brother Jackson made his NHL debut with the Flyers in 2021,[70] and he has most recently appeared with the Rockford IceHogs, the American Hockey League (AHL) affiliate of the Chicago Blackhawks.