I reviewed what Andrew Sihler's comparative grammar of Greek and Latin has to say about aspect and the augment, and here's my distillation of it.
Sihler regards the parent language (PIE) as distinguishing eventive verbs and stative verbs. Stative verbs did not have tense and eventually became the Greek perfect. The Greek pluperfect and future perfect are later developments.
As for eventive verbs, Sihler holds there is a distinction between durative (imperfective) verbs and punctual verbs (aorist). According to Sihler, punctual (aorist) verbs did not have tense.
As for durative (imperfective) verbs, Sihler holds that they are distinguished by tense. The default is the imperfect tense, and it has the default so-called "secondary" endings. The primary endings are actually derived from the secondary endings and they mark the durative verb as "non-past." Durative verbs with primary endings then became present-tense verbs.
At some point in Greek, holds Sihler, the augment, derived from some kind of adverb, began to be attached to imperfects and anterior aorists to mark them as past tense. Only the first verb of a narrative sequence needed to be marked for tense (augment) and mood, and it could be followed by augmentless secondary ending verbs. Eventually, the augment takes over the imperfect and aorists, and originally augmentless aorists either acquired the augment or got re-analyzed as historical presents.
So, if I understand Sihler correctly, the original Greek verbal system was primarily aspectual, but there was an unmarked past / marked non-past tense distinction for durative, eventive verbs. Then, the augment was added to mark imperfects and certain aorists for past tense, and then generalized for all aorists. The historical presents in this scheme derive from aorists that were unmarked for time. So tense was indeed marked in Greek verb system according to Sihler, but originally only for imperfective indicatives, where non-past duratives were marked with primarily endings and then, later, past duratives were marked with the augment.
Stephen
