Yes, every business needs to be recapitalized at some point. There is an entire industry that focuses on finding the non-public companies that OP mentions. It's called Private Equity and they buy these companies when they have come to the end of their natural cash flowing life.
The owners have either (at great pain) developed a sufficient capital base and succession plan to provide continuity (which would require significant ongoing investment in R&D out of cash flow in order to develop competitive products), or they must find a plan for the business. Going public solves this problem for them, so does selling privately. Neither option really eliminates the requirement to remain competitive.
Story after story involving a business getting bought by private ends with the PE 1) buying an ENORMOUS amount of debt based on the goodwill of the history of the company, 2) giving themselves ENORMOUS bonuses, and then 3) filing bankruptcy and liquidating. This is an entire AREA of business that should just not exist. It would seem that "investors" selling debt to these PE parasites would learn their lessons, but I expect that the even-bigger lesson is that they're all in on the grift together, and they're ALL making money somehow.
The owners have either (at great pain) developed a sufficient capital base and succession plan to provide continuity (which would require significant ongoing investment in R&D out of cash flow in order to develop competitive products), or they must find a plan for the business. Going public solves this problem for them, so does selling privately. Neither option really eliminates the requirement to remain competitive.