Basic chronology:
1942-1947: Betz is the best player in the world, but World War II limits the Majors she can play. Betz wins her last Major at the 1946 U.S. National Championships, beating Doris Hart in the final. In 1947, Betz states she's exploring the possibility of turning pro and is banned from amateur tennis. Betz is 19-2 against Hart at this point. Betz embarks on a pro career, most notably going 27-7 against Sarah Palfrey Cooke, the #2 player behind Betz before Cooke herself turned pro.
1948-1952 French Open: Margaret Osborne duPont, Louise Brough, and Doris Hart battle for supremacy at the top of women's tennis. Betz was 18-8 against duPont, 16-4 against Brough, and, as noted, 19-2 against Hart. Hart eventually emerges as the queen of women's amateur tennis, winning Wimbledon in 1951 and the French Open in 1952.
Wimbledon 1952-Wimbledon 1954: Maureen “Little Mo” Connolly takes the tennis world by storm, winning 8 straight Majors she plays (skipping the 1954 Australian Open). Connolly wins the Grand Slam in 1953, beating Doris Hart in the final of every Major except for the Australian Open, which Hart didn’t play. Hart is Connolly’s fiercest competition, starting out 3-0 against Little Mo before Connolly wins seven of their last nine matches (losing in Orange in 1952 and the Italian Open in 1953) to finish the H2H at 7-5.
July 1954: In the wake of the Wimbledon win, Connolly plans to turn pro. The plan is to have her play a series of matches against Betz as part of Jack Kramer’s Pro Tour, which Betz has dominated for years. But Connolly has her horse riding accident on July 20, 1954, knocking her out of the game for good.
August 1954-1955: With Connolly out of the game, Doris Hart returns to her position at the top of the women’s amateur game, winning back-to-back U.S. National Championships in 1954 and 1955.
1956: Hart turns pro and faces Betz twice, in Cleveland and Milwaukee. The Cleveland match is close, with Betz winning, 21-16, 19-21, 21-12. The Milwaukee match is more routine, with Betz winning, 6-3, 6-3. Betz finishes the H2H 21-2 against Hart.
So, to answer your question, Betz didn’t play Connolly because (1) Betz was banned from the amateur game when Connolly ruled the roost; and (2) Connolly had her horse riding accident in July 1954, just as she was about to play a series of pro matches against Betz.