Am I missing something in my preferences? When printing some articles from web pages, the text renders as highlightable while others render as images which I am not able to highlight. Using OCR and enhanced OCR does not help.
Can you tell us how you are creating the PDFs from the web pages? There are so many variables - from both a source and conversion perspective. If the text on a web page is an image (scan or photo) of text, that's how it will be in the PDF creation initially. You can run Acrobat's built-in OCR feature to convert the image-based PDFs to Text-included ones (with varying levels of accuracy). You mentioned that the OCR didn't help. Might the PDF have images and text and maybe handwritten content interspersed? That scenario can be quite challenging for the OCR to interpret. If you have the availability to do so, use the PDF conversion tool that is part of the Acrobat installation to convert from Web to PDF. I find it has the best chance to get the conversion right.
Any more details you can provide, or even sharing a file that has issues could help troubleshoot this.