I don’t think Renly was waiting. Rather he seems to have been moving carefully but consistently throughout 298 AC, and just didn’t forsee that Robert was going to drop dead at an early age.
I don’t have the sense that Renly started earlier than the others, in part because Renly’s really young (only 21 when AGOT begins in 298 AC) to start with, which means that he wasn’t in the game yet when Edric was born in 287 AC (being only 10 years old), or when Tommen was born in 291 (14 years old).