I don't want to dwell on how immaturely Paul came off with his claim of being the first politically engaged or anti-war Beatle. So just a couple more comments and then we can put the issue to rest.
Let's take Paul at his word and concede that he facilitated The Beatles' involvement in the peace movement. So he met with Bertrand Russell once (the news reports only mention one encounter anyway), became concerned about the Vietnam War, convinced his fellow Beatles of its repellent nature, and then what? How exactly did he follow through on his newfound political awareness? In what way did he try to meaningfully participate in pacifist activism? For doesn't that matter more? How you act upon principles and conviction when they've been awoken to great wrongdoing, etc. Now I'm not a Beatles historian but I don't doubt that Paul contributed to the peace movement in laudable ways. However, he's not an anti-war icon. And, of course, John is. For the cause, John wrote songs, arranged concerts, showed up at rallies, battled deportation, attended the Watergate hearings, and even staged "Bed-ins for Peace." He was a foot-soldier, though one with immense notoriety. It's an unavoidable part of his legacy. And it's something that Paul's will always lack, regardless of how passionate his, umm, animal rights activism is (no offense to these pursuits, but I put a much higher value on work that supports human life). So sure Paul*, you might have introduced John to the moral calamity that was the Vietnam War. But his reaction, a decade and a half of spirited anti-war championing, will always loom quite a bit larger.
*- I hate to sound so disparaging of Paul. But in this case, I think he exercised very suspect judgment.