The SAs may choose to send an LOA for any number of reasons that support the goals they have for their build-out of a well-rounded class, based on whatever portions of the application that interest them, at any point in the admissions cycle they choose.
Yes, LOAs do start appearing in August.
LOAs, of course, generally have one or more conditions that must be fulfilled before an appointment can be offered, such as obtaining a nomination, submitting an acceptable CFA, being fully qualified by DoDMERB, etc.
Do not drive yourself crazy looking for an LOA. Submit your best application as early as you can. Most particularly, do not look left or right, and wonder why another person got an LOA with what you perceive as lesser stats or weaker resumé, and you didn’t. See my bolded sentence. No applicant can see the totality of another candidate’s application, much less the entire applicant pool, in a comparative way.