The expression "to put someone/something through its paces" means to test something. It originally related to horses: a pace is like a trot, but the two legs on the same side of the horse move forward together. There are various speeds and styles of pace, and some can be uncomfortable on a particular horse, so it's important to test the horse's paces.
Both of the examples that you quoted are not typical, because the form is "he/she went through his/her paces"- the test is self-imposed. The first example is written by a sport writer, so you have to make allowances. *See Richard Winters' comment below.
Slaven Bilić was appointed head coach of West Bromwich Albion football club in June 2019. They were scheduled for a game with Middlesbrough football club on October 19th 2019 and all of the tickets were sold. Slaven Bilić went through his paces, showing how well he could manage the team. His efforts were successful: West Brom defeated Middlesbrough 1-nil, and they were later promoted to the premier league in July 2020.
I have no idea who Reed might be, or what she was doing.
Here are some conventional usages for a person and a thing: the first is Bilić again, but by a more literate sport writer. You can follow the links to get more information about what the sentence means.
West Ham manager Slaven Bilić put his side through their paces ahead of Swansea clash - Daily Mail, 2015
A new ground receiving station at the Cuban Meteorological Institute (INSMET) was put through its paces by a tremendous natural force - *UKRI, 2019*