Why "out" in "eat your heart out"?

I used the phrase the other day and it struck me as odd that out is needed. Wiktionary cites the following etymology of sorts:

Disputed. Three schools of thought exist:

From "This will eat your heart out.", suggesting that the recipient of the taunt will have their heart, the core of their being, eaten out with desire, bitterness, or pain.

From the 16th century "to eat one's own heart" (to suffer in silence from anguish or grief), possibly from the Bible "to eat one's own flesh" (to be lazy) The phrase "to eat one's heart out" appears as a formulaic phrase in the Iliad, meaning to experience extreme grief. (For instance, Iliad.24.128, many other locations.)

When used as the taunt "Eat your heart out, [someone]!" a suggestion that the recipient of the taunt "eat up" as much as they like. (From the same construction as "dance your heart out," etc.) Literally, similar to "have all you can eat!" Figuratively more akin to "experience me besting you."

The only one that seems relevant here is the last one which compares the phrase to dance your heart out. I am not familiar with that idiom and, according to this Ngram, it is far less common than eat your heart out:

The only other eat X out idiom I know of is the sexual meaning of eating someone out.

So, why out in eat your heart out? Why not just eat your heart? The out does seem to make a difference, but I don't know if that's just because I am used to the phrase. Is there an explanation for what the out confers to it?

Contrary to popular belief, 'eating one's heart out' does not necessarily have to mean eating to one's content.

It is a gloating expression, commonly uttered in a sneery tone. For example, you've just achieved an award, won something, or perhaps just done something commendable and you say it to someone who isn't at all happy with your newfound success.

eat your heart out

something that you say which means that you or someone you know can do something better than a person who is famous for doing that thing

'I've just made this coin magically disappear in just a flick of a finger. Eat your heart out, David Copperfield!'

When you are “eating your heart out”, it brings out a grotesque imagery of a heart being devoured, implying that your heart is being “eaten up” and crushed by some painful emotion, usually grief, or jealousy.

Looking at its origin, the earliest reference dates all the way back to 850 BC in Ancient Greece, where it has been found in Homer’s classical text: The Iliad.

In the story, Bellerophon, is described to be “eating his heart out” in grief when Artemis slayed his children. Since then, the notion that grief eats at the heart has been expanded to other emotions usually with negative connotations, for example, jealousy.

Regarding the question, my personal take on why the 'out' is used in eat your heart 'out' would be the emphasis on eating one's heart. Compare 'eat your heart' and 'eat your heart out'. The latter phrase definitely provides a more vivid and emotional imagery. According to thefreedictionary.com, out means "to the fullest extent or degree", and this shows the outburst of emotion that one is facing. Although both might mean the same literal thing, eating one's heart out seems to provide more emotional outburst to a higher degree; it is as if someone is literally removing the heart out of the body and devouring it in grief.

Taking a look at this particular type of phrase, there are some phrases that come across my mind too in the exact same format of x one's heart out, like for example, shout your lungs out.

Now I will try attempt to debunk the phrase of x one's heart out. Keep in mind that the meaning of eating one's heart out is to boast. I believe its meaning evolved to eating to one's content when people started to use the phrase during a served dinner. For example,

"There you go. Eat your heart out."

Technically, with the meaning of boasting, those who knew the actual meaning of the phrase would assume a light-hearted boast about the good quality of the cooking - perhaps even suggesting that one should be envious of such a talent of cooking good food. However, those without prior knowledge would assume naturally of course - to eat to one's content. Over the years, x your heart out would naturally evolve to mean do something to your heart's content, thus giving birth to the phrases dancing one's heart out.