Why don't index ETFs use leverage to offset their fees and eliminate tracking errors?

Leverage can be good to increase gains, but it amplifies movements of a stock. Also downward movements. Paired with the increased gains is also an increased risk, and that changes the investment vehicle. The goal of an index fund is to passively follow an index as closely as possible. Leverage could reduce tracking error on upward movement, but would increase the error when the index goes down: not only would the fund still have to pay back the loan (for debt-based leverage), but it would have to sell other assets at a low price to do so. Ouch.

There are other ways how a fund can reduce the impact of management fees. A very common way for funds using physical replication is securities lending: the ETF lends out some of its stock to short sellers for a fixed duration, after which they have to return the stock. The counterparty posts a collateral (such as cash) at the current value of the stock, and also pays a small premium. There still is some risk: if the counterparty defaults and the stock goes up by a lot, the fund is only left with the collateral+premium. However, this is a more manageable risk than depending directly on market movements.

In most cases, the income from securities lending goes to the management company, which (in theory) allows them to operate at a lower expense ratio. For Vanguard funds, the income generally flows back into the fund and can directly offset the TER, and can thus help achieve a real-world tracking difference of zero (or possibly even negative).

So the answer is: leverage is not generally used in index funds because better techniques such as securities lending exist. I'd also argue that funds in the 0.1x% TER regime are already so cheap (compared to market fluctuations and retail transaction costs) that no further optimizations are necessary. Of course, there are funds that explicitly use leverage as part of their investment strategy. But those typically have higher expense ratios (because they are more niche, and because managing them requires more effort, similar to active management). Those option contracts or loans don't come for free.