The main reason you cannot add a product to a membership is if it is not published - aka it's in draft mode.
To add a product to a membership, ensure that it is first published by going to Products - and then clicking on the 3 dots to open the product settings:
Now on the products edit screen - look at the top right of the screen and you should see a "Publish" link:
Click this button to publish your product. Then go back to your membership and you should find this product can be added successfully.