MOSFET as OE for 74HC165
Yes, the '165 is just a shift register. It was never intended to be a fully SPI-compatible device.

Using an N-channel MOSFET would convert its output into an open-collector (really open-drain) output with enable. If you go this route, you'll need to have a pullup resistor on the MISO line, because the MOSFET will only be able to pull it low. Also, the gate of the MOSFET is active-high, so you'll need to invert your SS signal to match. (Actually, I guess you can use the same signal that you're driving the '165's \$\text{SH}/\overline{\text{LD}}\$ input to drive the MOSFET as well, since you want it enabled while you're shifting.)

The '125 has an active-low enable, and its output is true 3-state, so this is actually the more direct approach, especially if you have more than one device you want to connect to MISO.