Liquidity is highly correlated to efficiency primarily because if an asset's price is not sampled during the time of a trade, it's price is unknown therefore inefficient. Past prices can be referenced, but they are not the price of the present. Prices of substitutes are even worse.
SPY is extremely efficient for an equity. If permitted, it could easily trade with much lower ticks and still have potential for a locked market.
An ideal exchange has no public restrictions on trade. This is not to say that private restrictions would need to be put in place for various reasons, but one would only do that if it were responsible for its own survival instead of being too big to fail.
In this market, trades would be approximately continuous for the largest securities and almost always locked because of continuous exchange fee competition with ever dropping minimum ticks.
A market that can provide continuous locked orders with infinite precision is perfectly efficient from the point of view of the investor because the value of one's holdings are always known.
In terms of the theory the Efficient Market Hypothesis this is irrelevant to the rational investor.
The rational investor will invest in the market at large of a given asset class, only increasing risk as wealth increases thus moving to more volatile asset classes when the volatility can be absorbed by excess wealth.
Here, liquidity is also helpful, the "two heads are better than one" way of thinking. The more invested in an asset class, the lower the class's variance and vice versa. Bonds, the least variant, dwarf equities which dwarf options, all in order of the least variance. Believe it or not, there was a day when bonds were almost as risky as equities.
For those concerned with EMH, liquidity is also believed to increase efficiency in some forms because liquidity is proportional to the number of individuals invested thus reducing the likelihood of an insufficient number of participants.
In the case of ETFs that do not perfectly track their underlying index less costs at all times between index changes, this is because they are forbidden from directly trading in the market on their own behalf.
If they were allowed and honest, the price would always be perfect and much more liquid than it otherwise should be since the combined frequency of all index members is much higher than any one alone.
If one was dishonest, it would try to defraud with higher or lower numbers; however, if insider trading were permitted, both would fail due to the prisoner's dilemma that there is no honor among thieves. Here, the market would detect the problem much sooner because the insiders would arbitrage the false price away.
Indirect internal efficiency
Taking emerging market ETFs as an example, the markets that those are invested into are heavily restricted, so their ETF to underlying price inefficiencies are more pronounced even though the ETFs are actually working to make those underlying markets more efficient because a price for them altogether is known.