Although in true history, the so-called Great Mountain clan was just a group of bandits occupying the Great Mountain, occasionally raiding cities, in the Eastern Country rich with kaidan culture, people loved to mythologize the rulers and then make demons of their enemies.
For instance, earth spiders were simply natives living in the mountains, and the two-faced Sukuna was a local hero who once rebelled against the authorities. Later on, even when famous figures slew bandit raiders, people would exaggerate their achievements by labeling those bandits as "ghosts" to highlight their martial prowess.
Since Yoshitsune killed Shuten-doji, a famous person, the latter's status naturally rose. After all, when you want to praise someone, the most direct and subtle way is to boast about the person they defeated.
