"My older brother is good at baseball."
Because 野球 is a noun, therefore doesn’t need the の. If the sentence was: « My brother is good at PLAYING baseball », you would’ve put 兄は野球をする and then added the の right after it, and finish the sentence with が上手です. My guess is that, when adding the character の right after a verb, the verb kinda acts like a noun, but I am no expert at this myself, just my guess.
兄は野球が上手です—> My older brother is good at baseball
兄は野球をするのが上手です—> My older brother is good at playing baseball
An answer using のが would’ve technically been correct, since we could tell from context you’re talking about the action of playing baseball. However, Duolingo doesn’t accept an answer with のが here, because it’s sometimes stupid like that.
It can be explained, and you nearly did. With 'baseball' being a strength (noun!), baseball becomes the grammatical subject of the sentence, rather than the person whose strength it is. Literally this translates to: "As for my brother, baseball is a strength/his forte".
The brother is indeed the topic - rather than the grammatical subject - of the sentence, but that's unrelated to the number of "は"s in a sentence (which there can be more than one of). Also, the number of Google hits is not an accurate standard for the correctness of a phrase.
As far as I know, the only time you get more than one は is when you're contrasting two different topics. That and starting with a set phrase like 「実は」Are there other patterns I'm forgetting?
I don't disagree with your second point, but number of Google hits is a good heuristic for determining whether or not a particular phrase is in common use, or if one phrase is more commonly used than another as in this case.