> I don't know at what point people were convinced that the push against immigration is some kind of billionaire plot

Much like most other leftist rhethotic, this belief is not based on any real logic. Basically, the entire process went like this:

1. Corporations want more immigration to suppress middle class wages and workers' rights

2. They run propaganda campaigns targeted at the left and their virtues ("the poor immigrants are suffering and need our help! only racist nazis disagree!")

3. Said leftists, desperate to virtue signal and avoid being seen as racist or xenophobic, immediately move to support the cause unconditionally

4. They see the world as purely black and white, good vs evil, so when they ask themselves "do billionaires want immigration?" the thought process goes "billionaires=bad and immigration=good, therefore billionaires hate immigration"