Why is Frank Burns portrayed as such a child in M.A.S.H?

To answer this, you have to consider the source material which is the film and the book that came before it. In the original material Frank Burns is a very different creature. While as inept as the character in the series, the film/book version is far stronger in his own personal beliefs and (unwarranted) confidence. Film-Burns goes so far as to blame others for the shortcomings of his lack of surgical knowledge and crushes a young orderly, blaming him for the death of a patient.

The book/film shows the danger created in wartime by drafting doctors to surgical duty who didn't necessarily have the ability yet still got promoted in the chain of command. It further shows the lengths that competent doctors went through to get these people thrown out of the army.

In transitioning to the series which wanted to still be drama mixed with comedy; albeit lighter fare so as to be accessible to 70s audiences; Frank Burns in his film/book state needed to undergo some changes.

• He needed to not be so much of a danger that he had to be removed
• He needed to be semi-likable

So, ratchet down the incompetence, ratchet up the child-like goofiness and you've still got "A" Frank Burns.

In the long run, the character didn't last the series and left under similar pretenses. Winchester was brought in spinning the film Frank in the other direction. High level of superiority and confidence, but removing most of the incompetence in exchange for cultural snobbery.

One last note on your comments.

Actors will tell you that villains/unlikable characters are far more fun to play. The fact that you find yourself brought to an emotional level over the character on a regular basis is a sign that the actor was succeeding at his work. Linville himself said he had a love/hate relation with the character. The actor himself was a very sweet, bright, well-liked gentleman, but really relished making Frank as conniving as he did.

Flanderization:

The act of taking a single (often minor) action or trait of a character within a work and exaggerating it more and more over time until it completely consumes the character. Most always, the trait/action becomes completely outlandish and it becomes their defining characteristic. Sitcoms and Sitcom characters are particularly susceptible to this, as are peripheral characters in shows with long runs.

Frank didn't start of as a walking punchline and at the start of the series is much more relatable, but was retrofitted with more ineptitude and scorn as the series dragged on.

Divergence from the original source material Characterization Marches On is pretty common - see also Radar O'Reilly whose cute factor was turned up to 10 as the series went on.

I believe the writers caught on that they had reduced the character to a walking punchline a bit too far and hence when Larry Linville left the replacement character played a similiar role, but had more depth ie. Major.Winchester

A lot of characters were Put On A Bus by the writers with Trapper being replaced with BJ Hunnicutt for less womanising and more back story, Henry Blake for Sherman Potter - less bumbling incompetence and more nuance

Frank was a miserable person. His wife was a cold fish who owned him lock, stock, and barrel. He was, at best, an average doctor, and he couldn't or wouldn't bring himself down to being human. He did show some flashes of humanity, but it was brief. And when Hot Lips Houlihan got engaged, tired of waiting, Burns just became a punchline, and it was the producer's fault. They would not let him grow.