The men mostly joining Renly’s cause and not Stannis’ shows that the common people could give two shits about rules established. Renly aiming to take the throne is literally another form of right of conquest. Might makes right is how Robert took the throne, and Renly was aiming to repeat.
You assuming I think Stannis would have been a worse king is something you imagined. Renly is a pussy, he wouldn’t have been able to handle the task. His idea of ruling revolves around the fun bits, but he’d flop at difficult decisions. His rule would fail just as Robert’s did.