HAVING MADE made many small changes to replace those elements in your sample sentence that have odd word choice or syntax and thus hinder natural flow, this is what I think a native writer of English would produce:
He clenched his fists, as though (he were) attempting to suppress some strong emotion.
This sentence is itself rather old-fashioned, and reminds me of the dramatic/gothic style in Dr. Jekyll, Dorian Gray and Sherlock Holmes. If you compare this with the original sentence, you could figure out where the writing of non-natives can differ from that of native English speakers.
As I am a non-native speaker of English myself, your question is indeed intriguing, and I think I am in a unique position (relative to native speakers) to provide not only personal insight but also some context as to how a non-native speaker learns the language very differently from native speakers, which leads to a different style of writing and possibly 'lack of natural flow.'
Please note that I am from India, which has been closely associated with Britain for over 300 years, and has therefore become very much an anglophone country, where English is the second language of a vast section of the population, an official language of all arms of the government, and the medium of higher education. Therefore we have had unique opportunities to learn the language at a higher standard from a relatively young age, with all its attendant benefits including the chance to improve our vocabulary and language by wide reading from an earlier age, and thus for more years.
The biggest difference between native and non-native learners is (usually) that the native learner learns the spoken language first, absorbing all its nuances from infancy, while the non-native learner starts with the written language, not speaking English at home (or even at school other than as part of learning grammar or reading aloud from textbooks) except in rare cases.
This difference affects 'natural flow' in the writing of non-native speakers as follows:
A. Even with extensive knowledge of book English and the ability to maintain correct grammar and usage, their writing can appear stiff, formal and 'stilted', lacking the nuance, 'ear for dialogue' and natural grace that native speakers easily bring to their written work.
B. The syntax and grammar of non-native speakers and writers is often strongly influenced by the syntax and grammar (and even idioms) of their native language(s), so that a lot of of artifacts/ minor errors/ odd usage can creep into their written work in a subtle way. In effect the person is likely to be unconsciously thinking in the native language and simultaneously translating into English, which affects both speech and writing.
I think this is sufficient to affect the natural narrative flow and create linguistic unease in the reader, thereby obstructing an immersive interaction with the written work.
[This is not a quotation; I have simply highlighted the most important part of my answer for easy reference.]
Members are invited to critique the text, writing style and 'natural flow' of this answer, treating it as a sample of written work from a well-educated and well-read non-native speaker of English for the purposes of this question, and provide your invaluable insights in comments.
(Note 2: The development of our spoken English is badly affected by the lack of opportunity to speak English at an early age, the dominant culture of the mother tongue, and the severe lack of exposure to native speakers; thus my own written English is much stronger than the spoken form, but that is a very different topic which is beyond the scope of this question.)