Because ETFs, unlike most other pooled investments, can be easily shorted, it is possible for institutional investors to take an arbitrage position that is long the underlying securities and short the ETF. The result is that in a well functioning market (where ETF prices are what they should be) these institutional investors would earn a risk-free profit equal to the fee amount.
How much is this amount, though? ETFs exist in a very competitive market. Not only do they compete with each other, but with index and mutual funds and with the possibility of constructing one's own portfolio of the underlying. ETF investors are very cost-conscious. As a result, ETF fees just barely cover their costs. Typically, ETF providers do not even do their own trading. They issue new shares only in exchange for a bundle of the underlying securities, so they have almost no costs. In order for an institutional investor to make money with the arbitrage you describe, they would need to be able to carry it out for less than the fees earned by the ETF. Unlike the ETF provider, these investors face borrowing and other shorting costs and limitations. As a result it is not profitable for them to attempt this. Note that even if they had no costs, their maximum upside would be a few basis points per year. Lots of low-risk investments do better than that.
I'd also like to address your question about what would happen if there was an ETF with exorbitant fees. Two things about your suggested outcome are incorrect.
If short sellers bid the price down significantly, then the shares would be cheap relative to their stream of future dividends and investors would again buy them. In a well-functioning market, you can't bid the price of something that clearly is backed by valuable underlying assets down to near zero, as you suggest in your question.
Notice that there are limitations to short selling. The more shares are short-sold, the more difficult it is to locate share to borrow for this purpose. At first brokers start charging additional fees. As borrowable shares become harder to find, they require that you obtain a "locate," which takes time and costs money. Finally they will not allow you to short at all. Unlimited short selling is not possible.
If there was an ETF that charged exorbitant fees, it would fail, but not because of short sellers. There is an even easier arbitrage strategy: Investors would buy the shares of the ETF (which would be cheaper than the value of the underlying because of the fees) and trade them back to the ETF provider in exchange for shares of the underlying. This would drain down the underlying asset pool until it was empty. In fact, it is this mechanism (the ability to trade ETF shares for shares of the underlying and vice versa) that keeps ETF prices fair (within a small tolerance) relative to the underlying indices.