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Many school districts with centralized school choice adopt strategyproof assignment mechanisms to relieve applicants of the need to strategize on the basis of beliefs about their own admissions chances. This paper shows that beliefs about admissions chances shape choice outcomes even when the assignment mechanism is strategyproof by influencing the way applicants search for schools, and that "smart matching platforms" that provide live feedback on admissions chances help applicants search more effectively. Motivated by a model in which applicants engage in costly search for schools and over-optimism can lead to under-search, we use data from a large-scale survey of choice participants in Chile to show that learning about schools is hard, that beliefs about admissions chances guide the decision to stop searching, and that applicants systematically underestimate non-placement risk. We then use RCT and RD research designs to evaluate scaled live feedback policies in the Chilean and New Haven choice systems. 22% of applicants submitting applications where risks of non-placement are high respond to warnings by adding schools to their lists, reducing non-placement risk by 58% and increasing test score value added at the schools where they enroll by 0.10 standard deviations. Reducing the strategic burden of school choice requires not just strategyproofness inside the centralized system, but also choice supports for the strategic decisions that inevitably remain outside of it.