Why can't quantum entanglement be used to send information faster than the speed of light?

I never understood why quantum entanglement couldn't be used to send information. The usual answer is that knowing what a wave function collapsed to doesn't change the fact that it does so randomly, and thus cannot transmit information. But that's not what I mean. Why can't you use the fact that the waves collapse to send information? When I asked my undergraduate physics professor about it (a long time ago), he just gave me a weird look and told me it can't work. Still, I've never understood why that is, and I've always been curious.

To be specific, I'll link a YouTube video explaining the quantum eraser experiment. You only need to watch the first minute and a half to understand what I am talking about:

https://www.youtube.com/watch?v=sQfSm6o-KlQ

Using the experimental set up in the video, I do not see what would prevent someone at M1 from sending information via Morse code to someone at M2. Specifically, have switching M1 on and off play the role of the on-off tone for Morse code.

This would be sending information at a rate that is nearly instantaneous, only being limited by the rate at which the Morse code representation of the information can be encoded/decoded.

Am I missing something?