'Green Screen Keyer' not letting me choose the keyed color

Press A on your keyboard to ensure the Select tool is chosen.

Make sure your clip in the timeline is selected.

In the keyer in the Inspector click on the square Sample Colour box.

Check that you have not got the Transform or any other tool active.

Yes, I have been caught out several times with that and other effects when I have not clicked the "Done" button in Transform.

I only remembered that after I posted my original suggestions.

This is a known bug in FCP 12.3. If you try to change it to whatever it defaults to when first applied, it doesn't work very well or at all. For example, using the Green Screen Keyer used to work better as a Luma Keyer than the Luma Keyer effect itself. It can't work at all for that any more. You can only key blue or green. You are no longer permitted to get creative with it.

I suggest giving Apple feedback via the Final Cut Pro > Provide Final Cut Pro Feedback... menu in FCP itself. If they get enough feedback, they may fix it within the next 2 years.