Why didn't Boeing produce its own regional jet?
Boeing did have a small regional jet called the Boeing 727. This plane was designed to operate at smaller airports, with independence from ground facilities as a selling feature. The best example is that the 727 had built-in stairs in the rear underbelly of the aircraft. This could be opened in flight until some hijacker known as DB Cooper jumped out to make his escape.

The 727-100C could have a combination of 54 passengers and  four cargo pallets of freight or up to 94 passengers in a mixed class with no freight.

Until it ended production at 1,832 aircraft, the 727 was the most popular passenger jet ever sold. Since that time, the most popular plane ever sold was the Boeing 737, which could have as few as 85 passengers with the 737-100.

Boeing met a definite need for airlines since it sold so many aircraft. A better question might be what could McDonnell-Douglas have done to meet airline needs and still be a viable competitor.

Two Class Seating Arrangement:




https://frequentlyflying.boardingarea.com/vintage-airline-seat-map-american-airlines-boeing-727-100-from-1977/