I'm coming from a Robinhood "day trading" mentality, so I'm used to public advantages being "priced in", more or less. It seems like many people pick index funds based on their expense ratio. I would think that this kind of thing is already priced into the index fund's value. Is this true?
To paraphrase the late great John Bogle, there are only two "free lunches" or "pure wins" in investing: One is diversification and the other is minimizing investment costs (fees). You really do come out ahead by choosing a lower-cost mutual fund or ETF.
So why is this advantage not arbitraged away by the market, as your trading intuition suggests? Free market arbitrage eliminates differential returns only to the level of the costs of arbitrage.
Mutual funds issue and redeem shares at NAV, while ETFs do so in exchange for a basket of securities. Thus, a mutual fund cannot "price in" anything but its holdings, and if its costs are uncompetitively high its assets under management will head toward zero. An ETF holding liquid securities will likewise track market value (for currently held quantities and prices) regardless of its expense ratio. In both cases, the costs will erode the holdings per share over time, of course, but a change in costs has no immediate price impact.
The fund costs (if not excessive) are accepted by retail investors because they have no better way of owning the index -- commissions to buy all the holdings individually would be prohibitive, but professional investors with negligible trading costs can perform the arbitrage. An exception would be an ETF holding illiquid securities, where low costs could result in a market premium because the ETF could become the most efficient way to invest in its holdings even for professionals. This is the one scenario where a fund would "price in" its expense ratio.
The other type of arbitrage would be shorting high-cost funds against a long position in their index to capture the erosion in holdings per share, but this is thwarted because a high fee is demanded to borrow such high-cost funds.
No, they are not. The expense ratio is not an indispensable part of the assets held by the ETF, like an expense, faced by company and impacting its share price. An ETF provider is always ready to redeem or subscribe shares at the NAV price (subject to a fee and minimum volume), so that it is highly unusual that an ETF would deviate a lot from its NAV. Even if an ETF provider were to increase tremendously their fees, most shareholders would redeem the shares at NAV thus making the ETF way smaller, but not necessarily impacting the price (barring any effect the ETF selling its holdings in the market has on the prices of the holdings).