How do I use my libary color swatch for Text and Color Overlay

Could you be over thinking it?  I have tried it with Windows 11 and Mac Sequoia, and I am able to simply click on a swatch regardless of what tool I have selected. 

You can also use the 'grab colours outside the Photoshop Window trick. 

Select the Eye Dropper tool.

Click anywhere over the document window and keep the mouse button held down.

Move the cursor anywhere including other apps, and the Library panel, and you see the foreground colour changing in real time.  Let go of the mouse button and the currently selected colour will stick.

OK, I tested using the Color Picker window, as in your screenshot, and that doesn't work.  Just click on the swatch and you are done.