I believe EMIYA is a more special case and can't be used as an example in this regard. He is fundamentally different from a normal Heroic Spirit. If you want to make the argument that all Counter Guardians are Heroic Spirits, then you also have to explain why EMIYA is the way that he is, since being used as a Counter Guardian is what made him regret his dream - if the version of him on the Throne is immutable, why does he remember and get bitter over
everything he's done?
One explanation is that it's due to his RM, but I think it's likely that being a Counter Guardian has fundamental differences.
Also also, here is official material from Fate/Complete Material III:
Speaking of which, the information (souls) making up the Servants return to the main body at the same time as the death of the Servant, and the main body can know, as records, about the actions of the Servants as if reading a book.
So the official answer is that Heroic Spirits on the Throne
absolutely can be and are changed by their summons, but there's a disconnect. They don't retain the memories, they just retain the information in a detached way as a record.
So it's entirely possible for someone on the Throne like Modred to read "Servant Mordred reconciled her issues with her father and liege after learning that said parent did not make her heir in order to protect her from the burden of kingship that was breaking Arthur's spirit" and reflect on that and lose her hatred. That said - and this is conjecture on my part - her hatred and betrayal are a part of her identity, so even if the main entity on the Throne changes, if a summoner had a high compatibility with the version of her who still had hatred in her heart, that version could probably still be summoned.
That
could be why EMIYA changes over time as well, but really I think the info we get on him between all the sources gives the impression that he has very vivid memories of the things that has happened while being a Guardian, so I still make the argument they are fundamentally different in that regard.