Why it could work
Bieber has a strong track record preceding him. He's won a Cy Young, a pitching triple crown, and a Gold Glove. He's also a two-time All-Star.
He had a strong season last year with the Guardians, posting an ERA of 2.88 in 31 starts. He also averaged 8.9 strikeouts per nine innings. He didn't strike out as many hitters this year, but he's averaged 7.3 per nine innings, which is still pretty good, so he'd help the Cardinals in terms of swing-and-miss pitching. For his career, he's averaged 10.2 strikeouts per nine.
He also doesn't walk many batters, averaging 2.5 per nine this year and just two per nine in his career. A career 3.16 FIP mark also isn't too shabby.