If I use the 74HC595 output shift register, and I don't use the latch pin, would it still work as shift register, like the 74HC164, that doesn't have that pin at all? The shift register is going to be used to reduce the output pins needed to drive the colums on a switch (pushbutton) matrix.
No. Both SRCLK and RCLK are edge-triggered, so if you hold RCLK at a fixed level, the outputs will never change.
You'd have to connect RCLK and SRCLK together in order to get the functionality you want, but note that this creates an extra clock of delay from the serial data input to the first output, as compared to the '164.