Cameo:
If you knew how little actual power the Board has you'd be embarrassed to have linked to that site. It is important for a charity to have well connected people on their Board. It isn't all that surprising at all to see execs from various industries on the Board of Governors. However the board has no authority over contracts, operations nor operational decisions.
Also the majority of the money raised is given to the International Federation of Red Cross (IFRC) for distribution during an International disaster response. The only money kept by the American Red Cross is for missions assigned to them by the IFRC.
The Red Cross by law must detail all their financials to the government and produce an annual report that can be reviewed by donors.
I think you are wise to do research before giving but without the lens of the paranoid.