To be more specific, let’s say a privately-held company XYZ decides to raise funds by going public. They go for an IPO, and a portion of the equity now becomes public. Let’s say the initial Share price is $10. Thanks to the IPO, the company XYZ has access to much more capital to fund its business operations. And the company works tirelessly to 1) make the company an attractive investment to new investors, and 2) to make shareholders happy by paying attractive dividends. I understand this.
Now, 20 years after the IPO if the company trades for $200 per share, does the company XYZ have access to the increased share price through any means? If so, how?
XYZ can make a secondary offering (create more outstanding shares) to raise additional capital. There is a limit to how many additional shares (how much dilution) the market can easily absorb, so more money can be raised when the stock price is higher.
XYZ can attract and retain employees with stock bonuses more effectively with the stock at $200 than at $10, and thus can save on other salary and benefit costs. (This can be thought of as similar to number 1 combined with using the offering proceeds to pay employees.) The incentive for employees with restricted stock grants to stay until vesting is also greater.
Note: Your scenario was that in the IPO "a portion of the equity now becomes public" and "the company XYZ has access to much more capital". This implies that the IPO was not (just) a sale of private shares but rather a creation of new shares with proceeds going to the company. A sale of private shares would not benefit the company itself, only those private shareholders. Likewise, the type of secondary offering that benefits a company with a highly valued stock is creation of new shares (sometimes termed as a sale of "treasury shares").