When investing into stocks, the currency used has little effect. Exchange rates definitely do influence how a stock develops, but their influence is a bit more subtle.
When you use bonds or other debt-based investments, they are typically bound to a particular currency. In the simplest form: you give me $100 now, I give you $105 back in a year (5% annual return). Clearly this is subject to exchange rate risk if you actually want to invest in Euros – e.g. you might invest €80 now but get back €74 or €90 depending on how exchange rates develop.
But stocks are different. The value of stocks is of course listed in a currency for bookkeeping purposes, but their value is more abstract. One way to conceptualize the value of a company is the expected value of all their future profits plus their current assets, with some discounting to account for risks and opportunity cost. The share represent a fraction of ownership, and thus a fraction of the company's value.
There is of course an interaction between earnings and exchange rates. If a fast food chain sells 10 million menus for 5 USD during a month, that will be 50M USD. Exchange rates move, so you might not get the “same” value after conversion to Euros. No larger company is isolated to a single nation, but has international supply chains and international income streams. If the value of USD compared to EUR plummets, this fast food chain might have “higher” earnings from their European subsidiaries, and might also have “lower” costs when paying for supplies in USD. A lower exchange rate can increase international competitiveness. Changes in exchange rates are random (for our purposes) and it's effectively random whether you'll end up a “winner” or a “loser”.
I'm using a lot of scare quotes here because a lot here depends on viewpoint.
Another reason why exchange rate have little effect lies in how the value of stocks develops. Generally, the true underlying value (independent from accounting currency) develops not additively, but multiplicatively. E.g. a company might have a 20% chance of closing on a $50M contract. A rational investor would value this chance at $10M, and add +$40M to the valuation when the contract closes.
However, the ability of the company to close on the contract has substantial second-order effects. They are less likely to go bankrupt, making future earnings more likely. They might be able to grow their business by 10%. They might be able to invest in R&D. These underlying changes in the ability of the company to earn money are best modelled as percentage changes in their value, which happens to be entirely independent of currency.
So yes, it is quite rational for an EU investor to invest on an USD-denominated index. The currency doesn't really matter, the percentage changes in value matter. And whether exchange rate fluctuations make you a winner or a loser is pretty random – for USD:EUR they will likely stay reasonably small anyway (compared to the fluctuations in value of underlying stocks). The exchange rates may also affect the underlying value of companies, thus amplifying or dampening the effect of fluctuations – but you're subject to those effects regardless of how you invest into stocks.
There is another reason why an EU investor should invest into non-EU stocks: geographic and political diversification. If the EU economy collapses, the Euro and investments in EU companies might become fairly worthless. But non-EU investments would be less affected, and would be even more “valuable” when converted into EUR. You describe the opposite scenario, that the USD drops in value and you “lose” on your investment. But those two are sides of the same coin. You cannot avoid risk, but you can reduce risk by spreading it out further.