Contact tracing has played a central role in COVID-19 control in many jurisdictions and is often used in conjunction with other measures such as travel restrictions and social distancing mandates. Contact tracing is made ineffective, however, by delays in testing, calling, and isolating. Even if delays are minimized, contact tracing can only prevent a fraction of onward transmissions from contacts. Without other measures in place, contact tracing alone is insufficient to prevent exponential growth in the number of cases. Even when used effectively with other measures, occasional bursts in call loads can overwhelm contact tracing systems and lead to a loss of control. We propose embracing approaches to COVID-19 control that broadly test individuals without symptoms, in whatever way is economically feasible – either with fast cheap tests that can be deployed widely, with pooled testing, or with screening of judiciously chosen groups of high-risk individuals. Only by ramping up testing of asymptomatic individuals can we avoid the inherent delays that limit the efficacy of contact tracing.