Olenna said it best - Renly's claim was pure treason.
The best chance the Realm stood after Robert's death would have been Stannis as King with Renly as his Hand. That would not be met with any opposition from the Reach, the Stormlands, the North, the Riverlands or the Eyrie, and with the Lannisters gone they could have patched things up with Dorne. That would have left the Westerlands isolated, and unless they planned on joining forces with the Ironborn who they despise, the Lannisters would be kept at bay.
Stannis would have been a frugal king who would have prioritised clearing the Iron Bank debt, and Renly would be the diplomat and champion of the common folk, which would counteract the effects of Stannis being a miserable grump with zero people skills. Alas, it was not to be.