Wow, I've been looking for something like this for a long time but all I could find was proprietary stuff or not-that-good free alternatives. I'm not being dramatic when I say I feel this is going to change my life, so thank you!
The challenge with Tesseract is that its OCR quality is (too) low for screenshots because screenshots are usually text on colored background. Tesseract is optimized for classical text document OCR. Of course, you can switch OCR engines with Sikuli (I did that before), but this requires touching the source code.