Question: "This idea is remarkably similar to the doctrine of election in Calvinist Protestantism, & raises the same questions, namely, "Why doesn't Amida make everyone trust him?" & "Where does our choice enter into the process?".
Question: "Then Amida N. doesn't make me trust him, he sends us all a message suggesting that we ought to, & those who do not respond have either failed to check their inboxes or can't make sense of the message. Is that it?"
It is because of Amida that one entrusts to Amida. The echo of Amida's Call in one's heart (shinjin/faith) is due to the power of the One who calls (Amida).
Just some people become open to this Call while others, don't. Your openness depends on how karmicaly matured you are in relation with this Call, this is why Rennyo Shonin emphasized the action of listening again and again to the Amida Dharma until we finally become open. Then, when we are open or karmicaly mature enough, we entrust. But this apparition of faith comes from Amida, just like the light which occurs in the room after we opened the window comes from the rays of light sent by the sun. The light which occurs in the room of our hearts (shinjin/faith), after we opened the window, it comes from the rays sent by Amida.
Opening the window does not create light (faith/shinjin). Light is not the product of the window nor the creation of the one who opens the window. Light comes only from the sun.
The truth that in fact, shinjin comes from Amida, even if it is us who listen to the Dharma about Amida is beyond our capacity to understand it with our limited and unenlightened minds. We are taught it is so, and we accept it, but we also experience this if we really entrust to Amida. Since I myself entrusted to Amida I always feel, even if I cannot understand it logically, that nothing related with my birth in the Pure Land really belongs to me, nor was the product of my efforts. For me it is simply so, everything comes from Amida, including shinjin and nembutsu.