The use of proper methodology and sourcing is more important than the author's education/credentials, but obviously you have to know what constitutes proper methodology and sourcing, which is where the education/credentials are useful. You can't really write (good) history without having studied it, because writing about history requires an understanding of the time period and issues you're writing about, as well as being familiar enough with the existing historiography on the subject to engage with it, which comes from reading and studying it.

When you're writing about history, you're still essentially following the scientific method (identify a question, form a hypothesis, do research to test that hypothesis, draw conclusions about your hypothesis based on the research you did) even if it's social science rather than hard science. You can't know what questions to ask, what hypotheses are plausible, and what research you need to do to test your hypothesis unless you have familiarity with the subject, regardless of whether that subject is chemistry or math or history.

The need for proper historiography and research methodology is why historians give more weight to peer-reviewed works of scholarship than non-peer-reviewed writing, because the peer-review process (at least in theory) means that work has been fact-checked and the use of sources and methodology has been validated by someone who is an expert on the subject. Credentialism/elitism can be a bad thing, but topical expertise and proper methodology are essential to good historical practice, and the way you develop the expertise and practical skills is through education.

I studied contemporary military history at university at both undergraduate and postgraduate levels. Personally, I read whatever interests me in a specific area, and it doesn’t really matter to me whether that person is an academic historian or not. Saying that I will always try and read something about the author. Many non-academic historians may come from a military background or journalism, but have extensive knowledge in a particular field. The book I am reading at the moment, Fighting for the Fatherland by David Stone, is written by a former British Army officer who had served for 35 years. The book is very well researched and he has used a good variety of sources. I have read a number of books over the years written by ex-military personnel, many autobiographies, but the historic ones are always well researched and written, some of which had personal experience of a conflict or a period of history.

So personally, I would say that if a history book is written by a reputable source, i.e. an academic or other, I would be happy to read it.

So in answer to your question, when I was at university we were given suggested reading lists for specific subjects, but they were never set in stone, and you were always encouraged to find other sources. We were never told to use just academic historians, but they had to be from a reputable source. When we learnt about research methodology, we were taught to be critical about how a book was written and also about the author. Certain internet sources weren’t seen as being reputable, so you would be marked down if you ever referenced from them.

I think it's fair to say it's less about who should write history and more whether it rises to the standard of academic history. Is it peer reviewed? Is it cited? How does it handle the problem of bias? But historians work from primary source material which is rarely written by historians. Serious original works of history rely on primary source materials. And while those can be newspaper articles or private diaries, sales figures or ship manifests, they can also be popular histories. Which are often both very helpful and often problematic since they may sensationalize or have a clear political agenda.

This varies by time period. A lot of medieval primary sources are "histories" by people who don't rise to the standards of modern historiography. You'll see this in this sub when people cite Roman or Anglo-Saxon sources. But I don't need to go to Pierre Burton to figure out what happened in the building of the railway in Canada.