Could someone take a look at this Tinkercad file?
Try this circuit: https://www.tinkercad.com/things/k4OdCknoxLM-shift-register-demo

I got it to work on my computer. Let me know if the link doesn't work.

The main difference compared to your design is that the MR pin has a pull up resistor so that when the button is not pressed, the pin reads 5V, but when it is pressed, the pin reads 0V. This is necessary because MR is active low.