The course L101 covers the issue of descriptivist and prescriptivist views of language and this is reflected in the Guardian article about a school discouraging the "use of slang" (the scare quotes reflect my view that some of the examples are creative uses of language rather than slang) https://www.theguardian.com/education/2021/sep/30/oh-my-days-linguists-lament-slang-ban-in-london-school
In particular, the expression "he cut his eyes at me" seems very expressive and eloquent and this seems to contradict the school's advice that following conventions will lead to eloquence. These kinds of expressions seem to be the kinds that are valued in literature where linguistic deviance is an important way of "making the world strange" (остранение) - for a brief summary of this see https://en.wikipedia.org/wiki/Defamiliarization and follow links to Russian formalists for more detail.
This is not to say I would always encourage these kinds of creative expressions (and perhaps it mainly seems creative to me because I had not come across it before). If clarity is required and the audience is older and would not know the expression, it should probably be avoided but this is different from saying it is not eloquent,
Ermm is a normal part of spoken language (see for example, Hultgren 2015: 126). In fact, people are often more intelligible if their speech contains some redundancy (and fillers like this are examples of redundancy). It would not normally be used in written texts but an awareness of the differences between spoken and written language is very useful in educational contexts.
Hultgren AK (2015) Exploring English Grammar Book 1: Grammar, context and meaning Milton Keynes: Open University