Undrafted out of high school, McGreevy enrolled at UC Santa Barbara where he pitched in 29 games as a reliever during his freshman season, posting a 5–1 record with six saves and was named first team All-Big West Conference and a freshman All-American by Baseball America.[3] As a sophomore, he went 2–0 with a 0.99 ERA in four starts before the season was cut short due to the coronavirus pandemic.[4][5] During his junior season, McGreevy became a first round prospect for the upcoming Major League Baseball draft.[6][7] He finished his junior season with a 9–2 record, a 2.92 ERA and 115 strikeouts against 11 walks over 101+2⁄3 innings.[8]
McGreevy was assigned to the Peoria Chiefs of the High-AMidwest League to begin the 2022 season.[13] After eight starts in which he went 3-1 with a 2.58 ERA over 45+1⁄3 innings, he was promoted to the Springfield Cardinals of the Double-ATexas League with whom he ended the season.[14] Over twenty starts with Springfield, McGreevy went 6-4 with a 4.64 ERA and 76 strikeouts over 99 innings.[15] He returned to Springfield to open the 2023 season and was quickly promoted to the Memphis Redbirds of the Triple-AInternational League. Over 27 starts between the two teams, McGreevy posted a 13-6 record, a 4.12 ERA, and 123 strikeouts over 153 innings.[16]
McGreevy was assigned to Memphis to open the 2024 season. In 20 starts before his promotion, he compiled a 5–7 record and 4.45 ERA with 96 strikeouts across 109+1⁄3 innings of work. On July 31, 2024, McGreevy was selected to the 40-man roster and promoted to the major leagues for the first time.[17] He made his MLB debut that night as the starting pitcher versus the Texas Rangers, giving up one earned run and striking out three batters in a 10-1 Cardinals win.[18]