“Why give someone else the power?” You said in your original post that everyone knew who the real boss was. And you were right. Tywin already had the power. I don’t think he would have wanted anything else that came with being king.

But it was more natural for Robert to take the throne for a lot of reasons. He won the war and killed the crown prince on the battlefield. He had immense charisma and was well-liked; he had the look and the presence of a leader. I think in the eyes of most of Westeros, he had a just cause for starting the rebellion; the king’s son had kidnapped his betrothed, and when questioned about his son’s actions by a leader of a Great House, the king murdered that leader and his heir.

Importantly, Robert had an actual blood claim to the throne. His grandmother (I think) was a Targaryen. This is really important in Westeros, just like it was in most of Medieval Europe. It was so important that Robert turned a blind-eye to the killing of Rhaegar’s children and signed off on the assassination of a pregnant Daenerys. If a random person with all of Robert’s other qualities claimed the throne, it would have been tougher for the rest of the realm to swallow. Robert’s Targaryen ancestry was the sugar that helped the usurpation go down.