Just the way he talked, he commanded respect. He was the pretty good at getting the rest of his family to do what wanted them to do.
Him sending Joffrey to bed at the small council meeting and advising Joffrey while he was sitting on the throne are classic examples of this. He did so by what he didn't say and how you could see others "reading between the lines".
Yeah, he was very good at governing, and he was played by such an awesome actor that I loved to watch him!
But the thing is, he couldn't have kept Joffrey in check for his entire reign, and probably for not much longer. Tywin was getting older and the little shit was just growing into manhood, at some point the little monster would have rebelled and either put some ass-kissing toady in as Hand, or tried to govern himself. And he probably have Tywin executed for treason, too, thats how medieval kings dealt with too-powerful advisors.
So while Tywin himself was way cool in his way, he was on a course that would inevitably end in disaster.
Ah, but Tywin had a strong sense of family loyalty and Joffrey did not - Tywin would have hesitated to kill his own grandson, but the reverse wasn't true.! So Joffrey would inevitably have ended up out of Tywin's control.
No, I adored Tywin, but he had put both Westeros and himself on a disastrous path.