Sidestick controllers for the 737 have been evaluated (at least by NASA), and there was some disagreement with this being too major a change from older airliners. The 737 with its high-bypass engines and design that has almost stood up to this day was a revolutionary aircraft already.
Airbus was entering their A320 into the market in 1987, a full 20 years later than the 737. Before the A320, if you wanted the most advanced large narrow-body, it was - technically, it was the 757, but since you didn't want to pay the cost and weight premium it commanded, and narrowbodies mostly serve shorter routes - it was the 737.
This meant Airbus needed a disruptive design with killer features to have any shot at the market. So they gave the A320 the most sophisticated fly-by-wire system, convenient sidestick control, and a ULD-supporting cargo hold. There was less concern about commonality as it was an all-new type anyway.
Both got what they wanted. Boeing got, with the 737, an airplane that was easier to retrain 707 pilots for. Airbus went from minimal market share to roughly equal footing with Boeing by differentiating their offering with the A320.
As time went on, both kept playing to their strengths. Boeing wanted to keep retraining pilots between their models easier, so they kept the yoke. There's still a lot of innovation in the background, but the pilot-perceived change is reduced when possible. Airbus kept developing their glass cockpits with further reduced pilot effort.