Not shown anything to indicate Kairos has which effect? If (as your sentence structure implies) you are saying that we're not shown other examples of it erasing memories, then that statement is easily disproved, so most likely you mean no other examples of burned-in memories emerging from reincarnated Kairos subjects. In that case, Hermes is the only example of it happening but also the only possible example of it occurring because all of the others known to have been subjected to it – Emet and Hyth – have not yet died and been reincarnated to test whether they also retain and dream of those memories.
What we do have is a lot of examples of people who have been reincarnated since the Final Days and were not Kairos subjects, and there is no widespread phenomenon of people having dreams and visions of their past lives. While that is not definitive proof against it ever happening, we don't have any proof in favour of it, and the game itself ascribes a different reason for Hermes' dream.
And if you have to preface something with "probably" then you're speculating. It's not fact.
That quote from Hermes is entirely about the memory he retained because of Kairos – a recurring dream he had over and over until it made sense, and then he got implanted with the Ascian memories that completed the picture besides this one retained snapshot.
He outright says in the scene you are quoting ("Her Children, One and All", cutscene #1) that being granted the memories of Fandaniel that he knew his visions were "the memories of Hermes, that he himself erased using the power of Kairos". Not all the rest of his memories, just that one segment.
Logically the second-hand "memories of Fandaniel" he received upon being turned Ascian would contain Emet's altered memory of how he recruited Hermes and there was a mishap with the memory-erasing device, and Amon would realise he is clearly holding the missing piece of that puzzle.