I don't think it has something to do with the transitivity of the verb after 好き. Indeed, there are cases where が and を are
seemingly interchangeable because of the transitive verb. For example, 彼 is the direct object of みなす in the latter sentence.
e.g.
警察は彼が犯人だとみなした。
警察は彼を犯人だとみなした。
However, this is not applied to your cases. ネコ can't be the direct object of 思う in the example 私は先生がネコを好きだと思います since the subject of 思う is 私. The speaker doesn't think about ネコ in this sentence, right? The direct object, i.e., what the speaker thinks about is actually 先生.
cf.
私は先生
がネコ好き(ねこずき=猫が好きな人)だと思います。
私は先生
をネコ好きだと思います。
Furthermore, いう is even not "to say" in your initial example. It's ~という you are familiar with.
Here's another example to show the transitivity of the verb is not the key.
バンザイ 君を好きでよかった
This is the first line of a song lyrics. As you can see, よかった doesn't take を as the object marker (it's not a verb, in the first place, though), thus, the transitivity of よかった can't be the reason why を is used here. If が is used instead of を, the meaning is ambiguous and it can be interpreted that 君 is the subject of 好き, i.e., the one who loves me/someone/something, so the writer would choose を consciously or unconsciously to avoid being misinterpreted.
I've coincidentally read 助詞「が」と「を」の置換性について mentioned in the page linked below, i.e., another thread of the same site you quoted.
日本語文法の初心者です :D In general, を is not used with 好き because 好き is a na-adjective: 彼は猫が好きです。 — correct 彼は猫を好きです。 — incorrect If this part is "embedded" as a clause, を is also
japanese.stackexchange.com
The writer Higashiyama points out that ~を好きだ is hardly used at the end of the sentence. He presumes that it's because が has a nuance of assertion, and ~を好きだと思います is acceptable because を is softer than が, so it fits the soft nuance of と思います.
~を好き is not so uncommon, by the way. Here's examples I found in a corpus.
父があれだけ狼に信頼された、狼を好きだった、それは、結局父が狼だったからだと思っているの。
夕子はお兄ちゃんを好きだった。
友達に協力を頼んだら彼女もその男性を好きだった、なんて展開になりがち。
お母さんのことを好きだったのか?
ボビーはきっと口で言ってる以上に、ジュリーを好きだったんだ。
それは、みんながニックを好きだからさ。
彼が兄さんの婚約者を好きだったし、今でも好きなのだと気づいて嫉妬する