Near-zero risk of what? If you mean "risk of decreasing in value", then checking and savings accounts aren't near-zero-risk either; cash tends to decrease in value over time due to inflation, and you're nearly guaranteed to lose money over time (though somewhat predictably). If you mean "risk of disappearing completely", then neither a savings account nor an index fund will do that.
A comment further up the thread asked for "very low risk"; the reply to that changed it to "near-zero risk".
Better to consider the type of risk you care about, and how much money you want associated with each type of risk. Checking accounts and index funds both make sense as part of an overall strategy. I don't know that savings accounts do, though, except perhaps as a purely organizational tool.