When a supervisor is not assertive, then in real terms, they are either aggressive or passive.
Aggressive supervisors use their positions of power and authority to belittle, intimidate, and control those who report to them.
As a result, their team members begin to resent them. Then their productivity begins to fall, and if the dictatorial behaviour of the supervisor continues, it will not be long before the organization begins experiencing a high employee turnover.
In other words, aggressive supervisors create an unfriendly work environment, which typically leads to unhappy workers, low productivity, and costly employee turnover.
In contrast, a passive supervisor is someone who is easy to walk over. Because the passive supervisor does not take things seriously, the employees will not take things seriously, and they will often do what they want despite the rules and regulations of the organization.
The primary reason for this is that the passive supervisor does not bother to enforce the rules. A passive supervisor usually avoids doing anything with their team members that might cause conflict, as they do not want to disrupt the workings of their department.
Hence they make needless concessions to demanding team members and even customers at the expense of the organization.
Unaware of the importance of assertiveness in leadership, many organizations have had to cope with tense and unfriendly work environment where productivity is suboptimal, and have lost some of their best employees, simply because of few aggressive supervisors.
Similarly, where supervisors are passive, resources of many organizations have unnecessarily been expended in forms of finance, materials, and man hours, and key deadlines have been missed.
These outcomes show how important it is for supervisors to be assertive. And considering the large number of direct reports most supervisors have, not being assertive usually has huge impact on overall productivity of the organization and eventually manifests as poor service quality.
But smart organizations are aware of the importance of assertiveness in leadership. In fact, they also know that it directly impacts employee engagement and in effect drives business outcomes.
They appreciate the fact that the primary determinant of an engaging and high-performing workplace is the supervisor, and that when supervisors exhibit high competence in team leadership, employees are more likely to be engaged. Hence they ensure their supervisors are truly competent. 
Competent supervisors are able to keep their team members from spiralling out of control. They always set and communicate clear boundaries and expectations. They hold everyone accountable, even when they don’t want to. When it is time for disciplinary action against a team member, competent supervisors have no problem with it. While they are not rude when reprimanding, they are not nice either. Without being aggressive, competent supervisors make everyone understand that their rules must be followed. They make their team members not just to comply, but to also commit. They are firm, fair, and consistent.
These abilities are driven by assertiveness skill. It may not, by itself, be a supervisor’s singular “secret weapon” trait, but it is a highly valuable characteristic.
Hence it is a necessary skill for every supervisor, manager, or team lead as it can be the difference between a productive team and an unproductive one.