Word for "No I in Team"

When a word is within a word, it is called a kangaroo word. Also known as: marsupial, or swallow word

Wikipedia says:

A kangaroo word is a word that contains letters of another word, in order, with the same meaning. For example: the word masculine contains the word male, which is a synonym of the first word; similarly, the word observe contains its synonym see.
The etymology of the phrase kangaroo word is from the fact that kangaroos carry their young (known as joeys) in a body pouch. Likewise, kangaroo words carry their joey words within themselves. Twin kangaroos are kangaroo words containing two joey words (for example: container features both tin and can). In contrast, an anti-kangaroo word is a word that contains its antonym; for example: covert carries overt, animosity carries amity

In Richard Lederer's, The Word Circus: A Letter-Perfect Book. 1997:

"Among the kangaroo words that yield the most joviality and joy are those that conceal multiple joeys. Let's now perambulate, ramble, and amble through an exhibit of this species. Open up a container and you get a can and a tin. When you have feasted, you ate and have fed. When you deteriorate, you rot and die. A routine is both rote and a rut. Brooding inside loneliness are both loss and oneness.

"A chariot is a car and a cart. A charitable foundation is both a fund and a font. Within the boundaries of a municipality reside city and unity, while a community includes county and city."

Thus harm in harmony could be said to be an anti-joey word and futility an anti-kangaroo word because it carries the word utility.

I also agree that there probably is no word to describe this.

How about calling it a "substring saying"?

Ex: You know that substring saying that goes, "There's no 'I' in team."? Well, have you heard, there is a me in team?

I would like to propose another option for the construction of the phrase. Kangaroo words certainly describe the unpacking portion of the construct, but I'd still hope we could come up with something to encompass the algorithmic usage described in the original question. Though I think substring saying is on the right path, I think besides my programmer and math buddies, it might not get much love.

Even after trying to stretch the definition of synecdoche, clitics, and tmesis (and then checking back in here and seeing the beauty of kangaroo word), I still wanted to stretch the inner-wordplay to an encompassing phrase. After trying out some other math terms with more generic application, I wrote down "Interset Reflection". Though interset has some maths usage, it also has a generic definition of "set among", and reflection speaks to the action needed to see the phrase meaning (you must look back and see that T-E-A-M does NOT have an I in it...) and the fact that thinking on the construction reveals the creative semanticism.

Thank you Dan for posting this and giving us all the chance to flex our linguistic muscles! Also, thanks to all the contributors for furthering this exercise!