In the previous era men either wore wigs or long hair in a queue (ponytail) and hair powder was used for both styling and grease control. The French Revolution made more high-maintenance and elaborate hairstyle unpopular in France while Duty on Hair Powder Act of 1795 curbed hair powder usage in England (supposedly Francis Russell, Duke of Bedford, started the trend with protesting hair powder by cutting his hair in a short natural style. This cut later became known as "Bedford Crop"). Note that the short natural style was mainly adapted by younger men, older men, lawyers, judges, servants of large households etc wore old-fashioned long hair or wigs well after the 1800's (for example here's James Madison still wearing a ponytail in 1821).
In addition Regency aesthetic was very much influenced by ancient Greek and Roman styles since ancient Greece and Republic Rome were seen as a ideals of democracy. For example Neoclassical architecture of the era frequently use elements like columns, women's clothing and hairstyles were copied from ancient marble sculptures (this particular painting depicted a particularly radical fashion in the French Directory style immediately after the French Revolution. The style quickly evolved into the Regency fashion familiar to us shortly after 1800).