The Sharpe ratio and the Sortino ratio are not under the control of the ETF managers, they will be equal (or very close) to the ratios for the Index that the ETF tracks. There is not much room for differentiation here.
To make their ETF attractive to customers, fund managers care about the tracking error between their fund and the index. They would like their ETF to track the Index as closely as possible. They will be monitoring this and taking steps to improve the tracking if necessary.
Similarly they also care about the bid/ask spread for the ETF and the discount/premium to NAV. Although they do not control this directly, they will be monitoring the situation and keeping in touch with the Authorized Participants and Market Makers to ensure the trading cost and NAV vs Price error are small, which will attract customers. If necessary they can encourage additional AP and MM to get involved.
Of course what all money managers care most is having a large AUM, since they are paid a percentage of AUM. They will try to market their fund to attract additional customers and their AUM. Perhaps their marketing materials will mention the Sharpe Ratio and the Sortino Ratio, but it is not a key metric as far as passive ETFs managers are concerned.