Why did they completely give up?

I think the act is a symbol of defiance. Stepping on the icon seems to them like flipping God off, and they're worried that this will permanently exile them. I think Kichijiro's storyline shows the interesting way that forgiveness is always there, while Rodriguez's storyline shows exactly what you're talking about: don't worry about the symbol, worry about the belief.

Maybe protestants wouldn't have been as worried about treading on the image, but these were Jesuits, after all.

These ideas - that faith is this weird, big, messy, glorious thing that is bigger than the perceptions of people - is, to me, central to the story. Rodriguez's struggle is his faith being taken through places he never dreamed it could go and finding out what God's place in his life *really* was.

I think Scorsese is trying to say what you're saying: it's not faithless to just step on the icon, especially when you're helping others by doing so. The plot (partially) concerns itself with the struggle it is for Rodriguez (and the others) to make that leap (of faith?) into a world where their belief is informed by, but no limited to, what they thought they "knew" about God.

I'm fairly convinced that neither Ferreira, nor Rodriguez, lost faith. I'm 99% sure Rodriguez remained a Christian his whole life. Why? The cross at the end. I think his wife put it there, but he wouldn't keep it at all if he was a "true" apostate. I think he actually converted his wife and family over to Christianity, they kept the relics they could, and worshipped in secret.

I do think Ferreira held on to more guilt over his apostasy.

My impression was that they were not allowed to leave. The narration talks about the Inquisitor demanding constant confessions and renunciations out of Rodriguez, he was obviously making the same demand of Ferreira with the book he was writing, and I think the insinuation is that the Inquisitor was making sure that the known narrative of these two priests was that they relented and saw "truth". He was using their stories for propaganda. If they left the country, they could change the narrative. So I think he didn't let them leave and I think he forced them to work at ferreting out Christian artifacts because that kept his propaganda going.

"So, putting aside the question should they/shouldn't they apostatise to save others, why did they seem to consider their cause completely lost once they stepped on the fumi-e? "

Once they stepped in, they would probably lose their priest´s authority from the local converts prespective, besides the word would likely spread that they would be left alone without a priest (being Rodrigues the last one)

Yes, I think the cross was placed by his wife; also Ferreira said he "doubted" after being questioned by Rodrigues. They both never seemed to be happy with their situation... who knows why they decided to remain, probably they realized they would manage to serve in another way.. in "silence".