So is there any definitive proof that holding a bank account with Chase, Citi, AMEX, etc... will help improve the odds for the borderline cases?

My personal experience with AMEX was that I cancelled my Green Card and applied for the Delta Gold Skymiles. This was when I had less than 2 years of credit history, lots of recent inquiries (about 7 in the last year), but a reasonably good FICO score of 740ish. I dumped about $20k into an AMEX savings account before applying... my Delta Skymiles was not auto-approved, but manually approved (but I did not did to call for reconsideration).

The thing I am trying to figure out is did the $20k AMEX savings helped the decision making process? Cause I was pretty sure I was on the borderline situation...

This brings to question if such a strategy would be effective. For the borderline cases, perhaps open a bank account with the card issuer, and dump in as much money as possible, just before applying for the CC. If having a banking relationship and the amount of money in the bank does affect the CC application, that would be good. After the CC is approved (or rejected), pullout all the money and just leave behind enough to avoid fees, or perhaps close the account altogether (assuming there is no account closure fee).