... its more like Barbarian is the only class stubborn enough, too focused on using his raw strength only to continue, despite there is no sign of progress right now ...
No, that's a thing of character, and characterisation, not class. If we can stubbornly try this despite the seeming difficulty, that's a character and personality choice, and should not be class locked, at all.
What a Barbarian
Should be able to do is contextually expend a rage when they fail the first time, to take the second check with advantage - that's one of the things rage is
For after all - advantage on strength checks.
