An intriguing tapestry of concepts, methodologies, and tools, Six Sigma is a disciplined, statistical-based, data-driven approach and continuous improvement methodology for eliminating defects in a product, process, or service. It was developed by Motorola and Bill Smith in the early 1980's based on the fundamentals of quality management. Let's delve deeper into the realm of Six Sigma consultants, illuminating their role in business enhancement and process improvement.

Six Sigma consultants, or Black Belts as they are often referred to, are highly-trained professionals who specialize in implementing Six Sigma methodologies within an organization. They are the warriors on the frontline, leading process improvement initiatives and mentoring Green Belts, the project-level workers in the Six Sigma hierarchy.

In the realm of economics, these consultants operate under the principle of Pareto Efficiency, a state of allocation of resources from which it is impossible to reallocate so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. Here, the resources are the business processes, and the allocation is the application of Six Sigma methodologies.

To understand the modus operandi of Six Sigma consultants, one must comprehend the DMAIC methodology - a five-phase approach to process improvement. This approach, an acronym for Define, Measure, Analyze, Improve, and Control, forms the core of their work.

In the Define phase, the consultant outlines the issue and establishes specific goals for the business process. This phase is akin to the scientific method, with the problem statement serving as a hypothesis. In the Measure phase, consultants quantify existing systems, establishing a numerical baseline from which they can work. This is reminiscent of mathematical proof, where evidence is required to substantiate a claim.

During the Analyze phase, the consultant scrutinizes the data collected, utilizing statistical methods to identify the root cause of the problem. This phase harkens back to Sherlock Holmes' crime-solving methodology, with its focus on deducing the culprit - in this case, the cause of process inefficiency or failure.

The Improve phase is where consultants develop, test, and implement solutions to the issue at hand. Here, the consultant plays the role of an engineer, reconfiguring processes to optimize efficiency. Finally, in the Control phase, consultants ensure that the new processes continue to work and improve. It's analogous to a gym trainer ensuring that his trainees maintain their new fitness levels.

Now, to answer the question of why Six Sigma consultants are essential, one must consider the role of continuous improvement in business. In a world of constant flux, businesses must adapt and improve to survive. In this context, Six Sigma consultants provide vital support. They bring an outside perspective, a rigorous methodology, and a deep understanding of business processes. They are instrumental in reducing variation, eliminating defects, and improving efficiency, thereby leading to cost savings and improved customer satisfaction.

Moreover, the influence of Six Sigma consultants extends beyond the business boundaries. As they drive process improvement, they also foster a culture of continuous improvement within the organization. By elevating the skill levels of in-house staff through training and mentoring, they contribute to the creation of a resilient, adaptable organization capable of weathering the winds of change.

To sum up, Six Sigma consultants are catalysts for change. They are the bridge between theory and practice, between problem and solution, between status quo and improvement. They bring to bear a potent combination of expertise, methodology, and perspective that can transform not just processes, but entire organizations. As such, they are an invaluable asset in the quest for business excellence.