Because much of the program is still present, even if some parts are absent or watered-down, or if some elements are added.
This is a topic that's too complex to address here.
Many countries have multiple Scouting organizations, that usually fit under a larger umbrella.
Germany has over 150 Scouting groups, some split by religious groups, some for labor unions, some for certain regions, some for . . . well, you name it. Norway has two primary groups. Having dozens and dozens of Scouting groups is common in Europe. In most of Central and South America, Catholic and Protestant Scouts never belong to the same Scouting organization.
Scouting is a worldwide movement, and the situation in the U.S. is somewhat rare. Countries with only one Scouting organization are usually either very small . . . or the group is run by the government.
If LDS Scouting left the B.S.A., I'm certain that the World Organization of the Scout Movement would recognize LDS Scouting as a separate Scout organization.