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With the [derog] I don't think the note is needed.

I'll use the xref stats here, since they are synonymous and easier to research.  Updated glosses (if any) should apply to this and the xref.

The [note] is adapted from wiktionary. いつも同じことを言う人をあざけて言うことば。

Looking at this both literally and in use, I don't think this is a proverb in Japanese, though we've glossed it with a proverb.  A more straightforward reading might be "a fool always flaunting their one bit of knowledge", or more precisely "the one bit of knowledge a fool won't stop repeating".  Maybe an allusion to a character in a proverb, but not a proverb itself.

Note in the ngrams:
馬鹿の一つ覚え	25831
馬鹿の一つ覚えのように	4007 <-- pretty high % of usage: "like a..."
Example google books sentence in context: repeating "kawaii kawaii" over and over:
「俺はお前の面倒なところに普通に腹が立って文句を言いたくなるが、やつは可愛い可愛いと馬鹿の一つ覚えのように言う」

A good tweet:
https://twitter.com/FIFI_Egypt/status/1619981303176392705
馬鹿の一つ覚えのように“多様性”って言葉を使いたがるけど、その多様性の意味を履き違えたらダメですよ。
Trans mine: You can eagerly spout the word "diversity" like an idiot with only a single idea in their head, but if you don't understand what diversity means, it's pointless.

Here someone mocks yet-another-post on proper covid etiquette:
https://twitter.com/hoshi_kujira/status/1472161382565318656
いつまで馬鹿の一つ覚えやってるんだ？
Trans mine: How long will you keep repeating this worn-out tripe?

(I found at least a couple very-similar covid-related uses)

The phrase seems to simultaneously mock both the speaker and the message. If it were a good idea, repetition wouldn't be so bad I imagine. Two cases I've seen are \
1. repeating the same "worn out" idea over a span of time
2. repeating an extremely simple idea (可愛い、美味しい、面白い) with no further analysis to add in a short period of time (contributing to "simplistic" in my note). 

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I think if [sK] on this あほう forms, we can simplify the readings?

"阿保の一つ覚え" has no ngrams, but definitely comes up in searches and on twitter (a natural orthography for a native typist). The older kanji form 阿房の一つ覚え is basically absent.