Regarding calling the council themselves, the thesis of the piece of that they didn't want to tip their hand like that, as any smell of a threat to Targaryen rule risked unifying them against the Southron Ambitions bloc. They wanted Rhaegar himself to call the council, revealing the extent of the Targ split, then put Robert forward as the unity candidate, with at least 4 and possibly 5 Great Houses backing him.
I'm not sure it's mentioned in the piece, but in the same Lady Dustin rant to Theon where we get the phrase Southron Ambitions, Dustin alleges that Rickard Stark's maester was behind his scheming:
Quote:
Maester Walys had been known as Walys Flowers. Flowers, Hill, Rivers, Snow … we give such names to baseborn children to mark them for what they are, but they are always quick to shed them. Walys Flowers had a Hightower girl for a mother … and an archmaester of the Citadel for a father, it was rumored. The grey rats are not as chaste as they would have us believe. Oldtown maesters are the worst of all. Once he forged his chain, his secret father and his friends wasted no time dispatching him to Winterfell to fill Lord Rickard's ears with poisoned words as sweet as honey. The Tully marriage was his notion, never doubt it...
There is some suggestion through the rest of the series that the maesters are anti-Targ, being opposed to magic. Marwyn even alleges that the Citadel had a hand in the Dance. If what Lady Dustin is saying is accurate, it seems unlikely that Walys's plan was to swap one Targ for another, rather than knocking the Targs off their perch entirely.