The HTML would include the IMG tag, the TITLE tag and the A HREF tag.
Okay. You must be running a different version of WordPress to me, because as I stated above the image upload via Write/Post does not insert [a href] link within an image tag.
WordPress generates the HTML for the image when it sends the thumbnail to the editor.
I don't know anything about an editor, but the software that drives WordPress produces semantically correct HTML for image tags because images are images - not links. There is a difference.
You can wrap your image within a hyperlink, as semantically correct HTML thus making the image 'clickable".
I understand you wish to cut down on valuable time, but seriously when you consider the time spent writing a post versus a microsend spent inserting additional HTML inside an image tag is hardly something to worry about.