Jared - welcome to the forum.  I don't think that the Read More tag is an adequate stand in for a legitimate internal link.  It is theoretically possible for you to write a function that would grab some piece of text and insert it there.  However, you're probably better off spending your time and effort on simply creating the links.

Our internal linking strategy is to have one link per 100 words of text.  The anchor text of the link needs to be a legitimate phrase that makes perfect sense in the context of the paragraph.  The page linked to needs to have the subject of the anchor text as it's primary content.

We don't link to the same place twice in a post or page.  We try to have relatively more links toward the top of the page and fewer links toward the bottom.  We seldom link from a single word, we generally link from a phrase.

I'd suggest that you be suspicious about advice that tells you to link from a single keyword.  That advice is likely to be old, and Google is favoring the "long tail" type of links.

No, it would be a complicated piece of code that would require logic to determine what to say and when to say it.  It would similar to writing a custom function for each post that would change the link for that post alone.

That one isn't for a beginner, nor the faint of heart.