Dealing with a stuttering child

I have always observed people who stutter from afar, until recently when one of the new pupils we admitted into our school happened to be a chronic stutterer. One particularly funny encounter I had with this child, I asked him a question, waited a few seconds for an answer, and then moved on to another thing when the answer was not forthcoming, only to discover he was still trying to string the words to the answer together after some minutes. Since then, I always try to ask him questions that won't require more than a few-word answers.

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As if that encounter is not telling enough, I recently discovered that my 3-year-old boy sometimes struggles with words, especially when he's about to start a sentence. Once he's able to get the first word across, the rest seems to flow easily. Initially, his mum opined that this was a result of trying to mimic a stutterer he interacts with (since his speech has been very smooth before now), perhaps in his school, but I didn't really buy into it. This could be something much more than just mimics.

According to westutter.org:

Stuttering most often begins between the ages of two and four, when children’s speech and language abilities are rapidly expanding. During this period of rapid growth, interactions between speech, language, and emotional development are hypothesized to influence the development and persistence of stuttering. As children produce longer and more complex sentences, their brain experiences higher demand. This cognitive demand can affect motor planning control necessary to produce fluent speech.

This opinion resonates with my boy as he falls within the age gap quoted and his ability to speak is yet to be fully developed. A certain part of me just hopes it doesn't get worse than what it is already while another part wishes there is something I can do to stem this tide.

According to investigations, this attribute is caused by a combination of genetic, environmental, and physiological factors. Some people outgrow this habit as they grow older while it persists or even gets worse in some others. The big question is, what can someone like me do to stop this development?

My wife always tries to shout at him to complete his words quickly in the hope that the condition can be arrested before it gets worse. However, I don't think this is the right way as the action is beginning to weigh on him psychologically. He will use his hand to cover his mouth while trying to talk or stay silent totally. I had to advise his mum before significant damage is inflicted on him psychologically. Now, I usually encourage him to talk and wait for him to say the first word, after which the rest becomes easier.

While my approach seems to be rejuvenating his normal self back to life, I am secretly concerned and eager to know if there is anything special I may need to do in order to ensure that things don't degenerate further. For now, I am quite happy that there seems to be no sign that it's getting worse.

Most of the solutions I found online mirror the approach I've taken so far, vis-à-vis speaking slowly and gently with him, allowing him to complete his words without hurrying him, not acting agitated or frustrated in reaction to his condition, etc.

Do you think there is anything special I need to do? I would like to borrow from someone else's experience. Please reach out to me in the comment section.

Thank you!