I’m sure you will eventually cover this somewhere down the road, but I have to ask. After the Red Wedding wouldn’t it have been more beneficial in the long term if Tywin had agreed to Joffrey’s wishes on purging Robb Starks former allies?

I know Tywin believes “if your enemies bend the knee you must help them to their feet otherwise nobody will bend the knee to you”. I for one believe in this doctrine as well but I feel that many of the former rebels may still feel very bitter towards the Lannisters for all the grievances they have suffered because of them. So even though they have since the Red Wedding reaffirmed their allegiance to the Crown, there’s no guarantee that most if not all of them will rebel again if given the chance.

Even though the Riverlords and Northern lords have been crushed at the Twins and now possess very limited military strength, they are still in a position to cause the Crown much trouble if opportunity arises. With Lannister power now weakening, many Riverlords especially in the current political climate would want to avenge themselves of the first wedding as well as any other transgressions by the Lannisters, which means they will not only rebel openly but also do it by rallying behind any of the Lannister’s enemies whether they be a Stark, Tully or any of the pretenders to the throne. 

I know it would have been more costly and even looked upon unfavorably but in the long run don’t you think that it would have been more sound to have done what Joffrey wanted and eliminated the houses that had followed the Starks in rebellion and most likely still secretly oppose the Crown, while also giving their seats to nobles who are loyal to the Lannisters?  

Well, let’s start with a very important factor in this decision: Robb Stark left the Riverlands part of his army behind when he went to the Twins, because he was planning on returning to the North: “aside from her brother Edmure’s modest retinue of friends, the lords of the Trident had remained to hold the riverlands while the king retook the north.”  These Riverlords have 11,000 soldiers between them. 

And while the Lannisters and Tyrells together have the manpower to destroy these remaining forces, their forces are split between many fronts: initially they have to retake Dragonstone and retake Storm’s End from Stannis, then the Tyrells send men to besiege Brightwater Keep, then the Tyrells send men to threaten King’s Landing if anything happens to Margaery, then the Golden Company lands, etc. And keep in mind, a lot of the Lannister forces demobilize  when Tywin’s body is sent back to the Rock.

So the best example of why the Lannisters didn’t do this is the second Siege of Riverrun, where poor Daven is trying to coordinate a military operation with only 1,500 Westermen under his command:

You’ve seen our numbers, Edmure. You’ve seen the ladders, the towers, the trebuchets, the rams. If I speak the command, my coz will bridge your moat and break your gate. Hundreds will die, most of them your own. Your former bannermen will make up the first wave of attackers, so you’ll start your day by killing the fathers and brothers of men who died for you at the Twins. The second wave will be Freys, I have no lack of those. My westermen will follow when your archers are short of arrows and your knights so weary they can hardly lift their blades.

Without the Riverlords, that first wave (which is what really demoralizes Edmure) doesn’t exist and instead the assault will have to go with Freys and Westermen leading the way and maybe the assault fails. 

Which brings me to the ultimate point: yes, on paper, the Lannisters and the Tyrells could completely destroy the armed strength of the Riverlands. But when you back someone into the corner, they fight like a trapped rat, and that pushes up the casualty rate. Just look at what happened at Dragonstone, where a token force of men killed a thousand Westermen. Now imagine that happening again and again in dozens of sieges across the Riverlands.