Where in the product datasheet does it tell me that this won't work with the built-in shiftIn function?
You will not find any reference to an Arduino in any chip's data sheet, especially as the chip pre dates the Arduino by at least 30 years. In the data sheet, the writing in the first column on the first page says data is clocked in on the positive transition of the clock.
The parameter that you are exceeding is the data set up time, the shift out function is not waiting long enough after the changing the output to the serial input of the chip before clocking the clock pin.