As we embark on an exploration into the intricate world of US Reverse Mortgage Companies, a voyage of rediscovery awaits us. The industry, like an elaborate tapestry, is woven with threads of economics, law, demography, and finance, each contributing its unique hue to the larger landscape. It is a world where the subtleties of numbers, percentages, and financial calculations are the keys to understanding the narrative. While keeping in mind our objective - to delve into the sector's key findings and insights, we must remember that this journey requires both intellectual curiosity and analytical prowess.

The first station of our exploration is to understand the protagonist of our narrative – the reverse mortgage. So, what exactly is this financial instrument? In simple terms, a reverse mortgage is a loan that allows homeowners who are age 62 or older to convert a portion of their home equity into cash while retaining homeownership. However, unlike a traditional mortgage where the homeowner makes payments to the lender, in a reverse mortgage, the lender makes payments to the homeowner. Fascinating, isn't it? The relevance of this financial product lies in its potential to serve as an additional income source for seniors, thereby enhancing their financial stability.

The next stage of our journey takes us deeper into the industry, peeling back layers to uncover the key players. The US reverse mortgage landscape is dominated by a handful of companies, among which American Advisors Group (AAG) stands out due to its vast market share. Other noteworthy names include Finance of America Reverse (FAR), Liberty Home Equity Solutions, and Reverse Mortgage Funding (RMF). The presence of these companies represents a robust and competitive industry, with each employing distinct strategies to carve out their niche and serve their target demographic.

While exploring the temporal dimension of our narrative, we need to revisit the 2008 financial crisis. The crisis led to a substantial regulatory overhaul of numerous financial sectors, including the reverse mortgage industry. In the aftermath, the Federal Housing Administration (FHA) tightened the regulations, thus making it harder to qualify for a reverse mortgage. This period could be seen as a watershed moment that profoundly shaped the industry's present state.

An examination of the industry's geographical spread reveals a significant concentration in states with higher populations of seniors. States like California, Florida, and Texas, with their substantial share of retirees, record a higher incidence of reverse mortgages. This geographical concentration suggests that demographic factors heavily influence the industry's operations and growth.

Now, let us delve into the labyrinth of ‘how?’ How does the industry operate? At its core, the industry's modus operandi is rooted in a complex interplay of financial calculations and risk assessments. The lenders assess the borrower's age, current interest rates, home value, and the FHA's mortgage limits to ascertain the loan amount. The risk assessment takes into account the potential for home value depreciation and the longevity of the borrower.

Finally, we embark on the most intriguing part of our journey – why? Why does this industry matter? The answer lies in its potential to address a critical societal challenge – financial instability among seniors. As the baby boomer generation ages, the need for diversified income sources becomes increasingly critical. Here, reverse mortgages could serve as a lifeline, providing financial flexibility and security.

In conclusion, the US reverse mortgage industry, while complex, is an ever-evolving landscape shaped by a myriad of financial, regulatory, demographic, and geographical factors. As we continue to grapple with the societal challenges posed by an aging population, the role of this industry is likely to become even more significant. Armed with the knowledge and insights gleaned from this exploration, we can better understand, engage with, and navigate this intricate world.