Why does Exchange ActiveSync not keep more than 'x' days of mail?
I don't think this is a setting on your phone or has anything to do with data transfer or memory consumption (at least as far as your device is concerned). I think this is a policy enforced on your Exchange account. Exchange ActiveSync allows account administrators to define policies on how email is delivered to and stored on your device.

Specifically, an email retention policy called Maximum Email Age specifies the maximum number of days of email items to synchronize to mobile devices. According to Microsoft, Exchange 2013 allows the following options for this policy (emphasis added):


  All, OneDay, ThreeDays, OneWeek, TwoWeeks, OneMonth


HTC has a PDF document that outlines the Supported Microsoft Exchange Policies in HTC Sense (Sense is your phone's stock UI) and this policy is supported by your device. It would seem as though your account's administrator has set this policy to only allow three days' worth of email storage. Justifications for this kind of policy vary; I recommend contacting your account administrator (or whoever operates the Exchange system you use).