Without Tywin showing up, Mace Tyrell doesn’t leave Tumbler’s Falls and the Tyrells never show up to the Battle of Blackwater – there’s a political deal on the table, but Mace needs Tywin to sign on the dotted line or he’s not going to move. (If you doubt me on this, see the first siege of Storm’s End)
And what people forget about the Blackwater is how important, and how fine-grained, timing was in that battle: Tyrion had fallen, the Gold Cloaks had broken, and Stannis’ men were prepping to get on boats and rafts and trans-ship to the northern bank when Tywin and the Tyrells arrived in the nick of time. Delay their arrival by even an hour or two, and Stannis takes the city.