However the exchange can easily correlate that these 5 tx, spread over 5 months, all used the same address and thus went to the same recipient.
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By contrast, if Alice provides a unique address to the exchange each time, there is no way to correlate those, and no privacy problem.
Silent payments only improve privacy in cases where you would have used the same address publicly already (like a donation address on a website). In any other case address reuse has always been bad for privacy.