What is the rationale for taking a profitable company that does not need additional funding public?

I think the main reason is to allow insiders who were compensated by stock (options) to cash those out. By creating an actual liquid market, it becomes a lot easier for people who hold the stock to sell it.

In general, direct listings allow more people to put money into the company, which should see its stock price rise. Which stock-holders like. Alternatively, it creates a more accurate view of the value of a company, which might make it easier to get loans.