Disclaimer, I work for a market-neutral fund, and have close friends high up in prop shops.
Presuming all strategies have a curve of diminishing marginal returns as assets under management increase, you would not expect any fund accepting outside money to have expected returns beating the market, but you would expect many of them to have a combination of correlation to the market and expected returns that would make them an attractive component in a basket of broad index ETFs and market-neutral funds. (Assuming risk-adjusted returns are the utility function being optimized. If variance is their preferred risk metric, this results in optimizing Sharpe ratio via mean-variance optimization, MVO.)
It's fair to assume that any fund manager is optimizing the sum of returns from their own personal investments in the fund plus fees from outside investors. They pick the place on the volume/risk-adjusted-returns curve that still keeps their fund attractive enough to outside investors, and maximizes their personal profits (personal returns plus fund fees).
If that optimal point on the volume/risk-adjusted returns curve for their particular strategy is at a point where risk-adjusted returns beat the market, then they maximize their returns by either never accepting outside funds (prop shops) or by not accepting additional funds and gradually buying out their investors (such as RenTech's famous Medallion fund).
So, (assuming diminishing marginal returns) it's not rational to simultaneously accept outside investment and beat the market on a risk-adjusted basis.
I suspect that many market-neutral funds could reliably beat the market on a risk-adjusted basis, but their volume/risk-adjusted-returns curve shape and their fee structures make it optimal for them to operate at a point on that curve where their expected returns are below the market.
Note that this rational self-interest optimization below market returns isn't bad for the investors. Under most fee structures, it ends up being close to maximizing total investor returns. Increasing percentage returns would mean kicking out some investors.
RenTech's Medallion Fund and many prop shops, and funds that are currently slowly buying out their investors seem to indicate there are at least some strategies where the optimal volume/returns trade-off is above market returns. You would expect all funds that are currently open to more outside investment to either be young and lacking capital or else have an optimal point on the volume/returns curve that is below market returns.
Note that as previously mentioned, a simple mean-variance optimization on a basket would allocate funds to both index ETFs and market-neutral funds returning a bit under the market on average. It's entirely possible that both fund investors and fund managers are being perfectly rational.
Of course, there are also plenty of people out there who fool themselves into thinking they know what they're doing. The world certainly isn't perfectly rational.
I'm just saying that in a perfectly rational world, assuming (1) utility function of risk-adjusted-returns (e.g. Sharpe ratio, resulting in mean-variance-optimization) (2) declining marginal returns on investment, you would expect all funds accepting outside investors (except for young funds desperate for money) to under-perform the market in expected returns.
Now, everyone talks about Sharpe ratio on the outside, but the particular risk models actually used internally by any fund are almost certainly not just variance of returns. I presume all funds simultaneously apply a mixture of commercially available risk models and internally developed risk models. Sharpe ratio is far from perfect, but it's a good least-common-denominator for discussion, and doesn't give away any secret sauce.
Side note: it would be rational for someone to take you up on your proposal and simply use index futures to take a highly leveraged position on your benchmark index. As long as they had enough money to make you whole in the case of bad tracking error and large downturns, their expected returns would be large. However, you wouldn't be very smart to take such an agreement instead of just getting leverage yourself. This demonstrates why risk-adjusted returns are usually more important than expected returns.