If I understand properly, you wanted to use the color picker for color stops of Gradient Tool.
I place here a solution because it looks very useful and not straightforward considering many questions in Google (and my 10 minutes self-updating research).
The workflow is:
Open Gradient tool panel
Window –> Gradient
or press Command/Control+F9, if you're on MacBook Pro etc you should use Fn+Command/Control+F9
Select the object with applied gradient
Click once on the color stop you want to change

Click on the Eyedropper tool or press I
Press Shift+Click on the color of one of the objects you want to apply to the color stop.
You're done.
Answer from Ilan on Stack ExchangeIf I understand properly, you wanted to use the color picker for color stops of Gradient Tool.
I place here a solution because it looks very useful and not straightforward considering many questions in Google (and my 10 minutes self-updating research).
The workflow is:
Open Gradient tool panel
Window –> Gradient
or press Command/Control+F9, if you're on MacBook Pro etc you should use Fn+Command/Control+F9
Select the object with applied gradient
Click once on the color stop you want to change

Click on the Eyedropper tool or press I
Press Shift+Click on the color of one of the objects you want to apply to the color stop.
You're done.
If you must, you can use the eye-dropper tool to load a color in the color well.
It's better to get in the habit of using swatches. Then you know you have right values throughout. For gradients you simply drag a swatch onto the swatch tab on the gradient slider.
Illustrator has a big leg up on Photoshop with global colors. If you check this option, you can change the color throughout your document by simply editing the swatch.
Videos
No matter what layer I’m on, I can’t use the gradient tool 😭 this assignment is already late and I can’t figure out what I’m doing wrong please help please even when I create a new layer above the others it still does not work at all
I'm working on an illustration composed of the linework, the colors (done with the brush) and shading done with purple at Multiple transparency at 30%. I saw a video where the guy creates a new layer with same transparency value of Multiple at 30%, clicks the gradient tool, draws a short line starting below the illustration (painted with several colors) and a beautiful shadow is cast on the illustration. He uses photoshop, I'm on Illustrator CS5. What am I missing here? I've been searching everywhere for hours and can't find anything that comes close to recreating this effect.
This is the video. He applies the effect at the 20:05 mark: https://www.youtube.com/watch?v=GhChRyWtMgM&t=1231s. I can't apply the Overlay effect he does after this either. Any help would be greatly appreciated. Starting to digitally color my scanned inks and these 2 steps would really elevate my game.
Thank you so much for your time and assistance.
Kind of late but hope my answer can be of help.
So the control that you are looking for is called the "Gradient annotator" can might be switched off accidentally by pressing the shortcut "cmd + opt + G" for mac. Not so sure about Windows.
To reactivate it simply press the shortcut again or go to View > Show Gradient Annotator and you should have the gradient controls back on.
All good answers (I'm working in CS 5.5) I've had this problem and also noticed that even though groups and compound paths may cause issues with gradients, they do not make gradients impossible. Sometimes you have to Ungroup, Release Compound Path (Object/Compound Path/...), then re-make the compound path (if there was one to begin with) and re-Group. Usually this will fix the problem for me.