I am not suggesting that you cannot convey both mood and tense information with the same pattern. I agree with that, and I already made that point in my English imperative example. I also agree that moods can restrict which tenses you can express, sometimes restricting it to only one possible tense (as with Turkish inferential).
The point I am making, is that by the definition of mood, "inferential" simply has to be a mood. The point of using it is to suggest a particular relationship with reality ("I didn't see this, but I heard it second-hand"). That's modality, i.e. mood. It also happens to restrict the temporality of the verb to the past.
> It clearly transmits the essential time information and also adds mood no-confirm structure on top.
What you seem to be referring to here is the actual vocal pattern that you attach to a verb root to signify gossip. Of course, word endings can convey both tense and mood, just as they can convey both gender and number. But they are still separate concepts.