You sure showed me right now When making shit up doesn't work, making a public spectacle ignoring the person one couldn't argue against will sure elevate their argument, it is known.
It's more because Varian could keep the people that are even more idiotic than him in check, like @Mardux said. Anduin is capable of spending a second about thinking about what's going on, but he lets things like Stormheim or Silithus happen, because some people in the Alliance don't give a shit about his peacemongering to the point of being willing to outright break his orders if it means they can attack some Horde. And since he's only all talk about said peacemongering, he then ignores that. Varian never let such a thing happen in the first place and even in case something like that had happened under his rule, chances are that he'd punish those guilty.
Because when it comes to Sylvanas' own experiences, the mouse either ran away or outright died, depending on the front. Varian's repeated banging of his head against the wall happened on the Ashenvale front, where he was locked in an eternal stalemate with the thickheaded-ness equal he found in Garrosh.
Also, while Anduin may be more wary, I'd disagree with the notion that he's not the type to dash into the fray altogether. Because that's precisely the thing he did at Lordaeron. It could be argued that it's because emotions were high after Teldrassil, but that would only give as explanation as to why, it wouldn't change that it was him dashing.
Because dash he did, without anything resembling an actual plan. Even putting Alliance's ginormous plot-armor at Lordaeron aside, the reason why they weren't all turned to sludge because of his unpreparedness there was Jaina's conveniently timed arrival that no one in the Alliance even knew about.