Does Proxy Hedging Fully Eliminate the Need for Delta Hedging on the Underlying Asset?

No, proxy hedging does not fully eliminate the need for delta hedging on the underlying asset. It only mitigates the directional risk to the extent of the correlation.

The residual risk (basis risk) still requires the market maker to maintain a delta hedge on the illiquid underlying, even if the size and frequency of rebalancing are reduced.