The Radical Republicans did pass most of their Reconstruction program by overriding the Presidential veto of Andrew Johnson, so the numbers were indeed there. The real question is whether Lincoln could have convinced the Radicals to go along with his program rather than push their own. As you say, we will never know the answer to that.
I suspect that the real reason that the Radicals had the power that they did is that the prevailing sentiment in the population of the North was to punish the seceding states. I’m not a historian, so I could be wrong about that, but I don’t think you’d have so many Radical Republicans elected in the House without such a widespread sentiment.
Lincoln also had the advantage of not only being a Repulican (Johnson was a Democrat) but also being a much better politician who was able to get along with just about anyone. Those two alone would have given him influence over Congress that Johnson never had.
I suspect that the real reason that the Radicals had the power that they did is that the prevailing sentiment in the population of the North was to punish the seceding states.
I think you are right in that there was a strong sentiment for vengeance, but was that the prevailing sentiment? I tend to doubt that. If it were I think that the rest of the country would not have been so quick to accept the South back into government on an equal footing as quickly as they did.