Prior to playing in the United States, Büscher competed for both VfL Bochum and SC Preußen Münster. He was also a member of the U-16 and U-18 German national teams. Named the top midfield player in the U-19 German Bundesliga in 2011-12. In 2014, Büscher joined the Syracuse Orange men's soccer program. During his freshman year, Büscher became an immediate starter for the Orange. There, he earned All-ACC Second Team honors and was named to the ACC All-Freshman Team. Büscher tallied his first collegiate assist on 27 September 2014 in a 1–0 win against the second-ranked Virginia. The assist to Skylar Thomas, gave the Orange a 1–0 win against Virginia. Throughout his freshman season, Büscher earned a reputation of being a set piece specialist. His assists in subsequent matches against Albany, Bucknell, Duke and Georgetown led to three game winning matches. Büscher returned to Syracuse for his sophomore season where he remain a key cog in the Orange lineup. Büscher lead the team in assists and was second on the team in goals scored, en route to the program's first-ever college cup run. Büscher played in and started in all 25 matches the Orange played in 2015.
D.C. United
Büscher was selected 11th overall in the 2016 MLS SuperDraft by D.C. United. He made his United debut on 23 February 2016, in a CONCACAF Champions League quarterfinal match against Queretaro F.C.. On 1 March 2016, he scored his first professional goal in the return leg of the Champions League against Queretaro, in a 1-1 draw. Büscher made his MLS debut on 6 March 2016 in a 1–4 loss at LA Galaxy, coming on for Luciano Acosta in the 75th minute. He scored his first MLS goal against Orlando City SC for D.C. United on 24 September 2016. On August 14, 2017, Büscher was loaned to the USL side, Rochester Rhinos for the remainder of the 2017 USL season. On 28 November, 2017 his contract option with United was declined.
On 13 February 2019, Büscher signed with Canadian Premier League club Cavalry FC for the team and league's inaugural season. He scored his first goal for Cavalry FC on May 22, 2019 versus Pacific FC in the Canadian Championship. Büscher finished the season with six goals and four assists in all competitions, helping Cavalry finish first in both the spring and fall seasons and qualifying for the 2019 Canadian Premier League Finals. Büscher was praised for his performance and listed as one of the best midfielders, and players, in the league.
TuS Haltern
On 1 February 2020, Büscher returned to Germany, signing with Regionalliga side TuS Haltern.
