Tywin and Co. aren't going to twiddle their thumbs waiting for Robb to gather the combined forces of the whole North and the powers that be. If Robb had his druthers, he would have done just that.
Robb was in no position to haggle. He can whinge all he wants about how the Late Lord Frey must needs pay his dues to THE KING IN THE NORF, but by the end of the day it's Lord Walder's say on who gets the Frey's alliance and how.
First off, the Vale has virtually no reason to see why they should join in the fray when they're doing pretty good so far as they stay neutral. Secondly, Lysa is Littlefinger's puppet. It's his word that will move Vale, not hers.
And did you think Tyrion won over the wildlings by singing tales of their valour and daring? Even if you found some way to work them to your side, the lords in Vale will certainly find that insulting, among other things.
If Balon actually listens to you, then you've managed the impossible. Otherwise... no. Balon does not give two shits about his son, and he certainly does not want to listen to some greenlander lord who calls himself king. You'll only be wasting your breath on an idiot who'll be Abel'ed soon enough, anyway.
And therein lies your first fatal mistake. You've handed Tywin Lannister the trap in a silver platter, and he didn't even have to work for it. You might as well scribble "THE HIGH GROUND IS A LIE" in a red marker all over the message while you're at it. So while you and your host march up to Tywin's battlefield of choice, marvelling at your own cunning, Tywin will work the other way around and use your position to his advantage.
Assuming you, by some miracle, have managed to amass this amount of manpower in such a short amount of time, you'll also still have to encounter Stannis Baratheon at Blackwater Bay. Renly is assassinated at this point, and the Martells will be happy to aid you in your Lannistermination endeavours anyway. Sparing Tommen and Myrcella will prove to be detrimental in the long run---Baratheon loyalists will plot to bring them to power again. And Cersei does not have the option to take the black. She's a woman.
That portion of your plan is completely and utterly senseless. Not only will you sound insane trying to prove Jon's Targaryen heritage, you also antagonise everyone involved in overthrowing the Targaryens by spitting on your aunt, uncle, and grandfather's (and by some extent your father as well) graves.
Are you serious? Stannis doesn't need Melisandre's fires to know this is a trap. And arrows? Jesus Christ, even Patchface can come up with a better strategy. Remind me again how you managed to defeat Tywin Lannister, of all things?
You've completely forgotten about Balon Greyjoy throughout all of this and allowed him to do fuck to whatever he wants to do. By this point the North is screwed, Theon or no Theon.
Cersei Lannister's starting to look like a master schemer now, and it's a terrifying thought.
You've failed to account for Dany, Aegon, and Euron. Granted, you did specifically state that it was TWotFK you mean to win, so I suppose I can't fault you for that. But by seating yourself om the Iron Throne, you've effectively made yourself a target of all three. And there's also the White Walkers you didn't consider. Without Stannis to send them aid, the North's doom is sealed. But hey, details.