Do actively managed ETFs help the economy in any way, and if so, how? Listening to the ARK Invest podcast, I've heard several times ETF managers saying that they have a positive impact on the economy because by choosing the most promising companies to invest in, they allow capital to be located to companies that deserve it. While I do get the argument for direct investment such as VC or PE, I don't see well how an ETF manager buying stocks in a company helps the company or some other element of the economy.
Do actively managed ETFs help the economy in any way, and if so, how?
Yes they do in in a several ways. The main ones are:

Trading (not just of the people who actively manage the EFT but generally) helps price discovery. Price discovery is an important role of stock market. It is socially beneficial for companies to be appropriately priced given their fundamentals and all other avaiable information as that helps divert capital to the right companies. Even if traders do not buy shares from company directly the market price of a share affects how much capital company get by issuing new shares. Furthermore, ETF are also not just for stocks but also commodities or exchange rates, where they help to correctly price those commodities or exchanges - which is again socially very valuable activity. Quick price discovery is one an important characteristics of well functioning capital markets. In turn empirically well-functioning capital markets have first order impact on economic growth  (have a look at Levine, 1997 or Levine 2005).

Trading improves market liquidity. It makes sure that whenever you want to liquidate stocks etc. you hold there is some buyer. For example, it would be much harder for people to save  by investing in stock market if it would be difficult to sell stocks you own when you need to. Having low liquidity can thus deter investment. In fact liquidity is so important that it is commonly listed in textbooks as one of the major determinants of asset demand (see Mishkin &amp; Eakins: Financial Markets and Institutions ch 4). Consequently, even ignoring anything else just the fact that there is someone trading and creating liquidity in itself creates value and is beneficial for the economy.