TL;DR is there are some divisions and dramas inside the Golden Company's leadership caused by the sudden onset of peace and the new context of Westeros. Lord Arryn decides to take up my prior offer of service from the Golden Company, but is unwilling to take in too many foreigners, especially since his main priority is making sure the Vale is safe and defended against the fanaticals and madmen across the Narrow Sea. His contract thus appeals to the Westerosi core of the Golden Company, knights and lancers and such, but excludes the Essosi and other "strange" elements, including lowly Westerosi who have grown used to the climate and/or comfort of Starfall or Sunspear. Tensions between the Company leadership come to a head with the death (assassination?) of Corlys Velaryon, whose death precipitates a succession crisis in Starfall: there's his infant daughter by his estranged (divorced?) wife and there's his infant son by Ashara Dayne. Rhaegar and the Martells chose Ashara's son, and generally the people at Starfall want Ashara in power/at the head, whereas the ones headed to the Vale are the ones most attached to the Blackfyre cause. Either to seal the deal and secure the loyalty of this professional army, or for certain other reasons, Lord Jon Arryn ends up marrying the widow of Starfall and thus normalising the settlement of the Golden Company in the Vale.
Rhaegar is not necessarily shut out of the story but also, in my mind, not an integral part. Since the timeskip is all about the two sides coming together I'd rather not have Rhaegar being the only one with agency or ownership on the storyline, if that makes sense.