The Alpha Legion's ulitmate purpose?
If you read Legion: Secrets and Lies it shows that the Alpha Legion likes playing the shell game with their Primarchs. They also use plastic surgery to make their marines look like them. It's also inferred that they are the galaxies smallest Primarchs since they cans blend in with their marines so well.
Supposedly one died in a duel with Roboute Guilliman, but who knows. IMHO the duel and the battle which led to it is so uncharacteristic of the Alpha Legion's tactics and so very fitting of the Ultramarines it smacks of being a setup. Especially considering that it was the Alphas who decided it's terms. They were defending, but up until that point they were doing everything they could to avoids a stand up fight, there was no reason to entrench and face a superior force head one. The Ultramarines did not outmaneuver them. Either Alpharius/Omegon got too cocky which goes directly against their character and thought they could beat the Ultramarines at their own game. Or some other marine took the fall for the Primarchs to satiate Roboute after inflicting enough losses on the Ultramarines to have them fall to an unknown xeno threat. Or maybe it was just to show the Ultramarines the superiority of covert tactics hoping they would adopt them to fight chaos since the whole Heresy thing didn't work out. Who knows.
Characterizing the Alphas as closet loyalists is incorrect IMHO. They're more antichaos than pro human. After watching the little slideshow they agreed to help engineer Horus's victory with the idea he would annihilate the human race in a giant guilt trip this depriving chaos of power. In the end they sided either with xenos or fell to chaos. They very much are the enemies of the Emperor at this point.