Chapter 23, where the letter Kiritsugu left behind for Saber was revealed, you can clearly see the grief and desperation that has gripped the man, to the point where he was actually putting his faith in a Heroic Spirit, an entity which he had despised for the better portion of his life due to them (from his point of view) being no better than common murderers.
Save them Saber. Please save my children. I have bet everything on you and Shirou to make things right where I have failed, where I have gambled everything on a lie and gained nothing but destruction. Every other attempt I have had at trying to make someone close to me happy and live has ended only in death, violence, and sending them on the path of ruin.
Please give me a reason to believe that there is worth in heroes one last time.
The 'blink and you'll miss it' mention about some of Archer's previous summons as a Counter Guardian. He's had to kill Sakuramultiple times.
Chapter 34 has Rin pretty much start to break down after Luvia and Princess!Medea help her realize that Archer's attitude, different Reality Marble, and lack of a recorded Legend all point to him being a Counter Guardian.
What makes it worse, is that because she's unaware that this Shirou is not her timeline's Shirou, it makes her believe that all her efforts to get curb Shirou's distortion were for naught.