It will sound more natural when "that" is added after believe.
Does it? I think omitting the "that" sounds perfectly fine.
That usage is somewhat dated according to writing style guides. It works but it's verbose without benefit.
cough YG cough
cough CUBE cough
음 실제로는 믿는다고는 얘기하지 않지 않나요
보통 -라고 생각한다거나 -라고 본다고 말할 텐데
a very useful sentence
Where does the 고 comes from?
The grammar rule is 저는 X다고 믿다 = I believe X, where X is a declarative sentence not ending in 이다. See https://www.duolingo.com/skill/ko/Quotation/tips-and-notes