Actually, I just got the Claymore on my Vanguard (and swapped it out for the Katana to complete that mission anyway), so it's not that. Killing speed is good pretty much right from the get-go; what improves as you level up is mostly your durability, since your Charge gives you more and more Barrier, and you can use the Charge more and more often. You can get sloppy (or charge into a hot zone, if you're careful) at higher levels, whereas at low levels you're still ripping stuff down really fast, but you have to be very, very careful about only charging into places where you can kill two enemies and then take cover until it recharges.
In fairness, it does sort of depend on what you're fighting, with regard to fighting through protection. A Vanguard outpaces an Infiltrator versus mercs and non-Harbinger Collectors, because Shotguns are better at fighting Shields and Barriers than Sniper Rifles, and they're both fairly equal at taking down health (the melee supplement tipping the scales in the Vanguard's favour). It definitely helps to have AP Ammo if you're going to be fighting Vorcha, Krogan, or mechs, and obviously now you've also got the Eviscerator to choose if you know Armour is going to be a concern. But on the whole, I'd still say Vanguard kills faster. Reducing enemy spawns might be another factor, but even in areas with set spawns I found I got through them quicker using the Vanguard (and I was already moving fairly quickly on an Infiltrator since it was my second go-around on Insanity).
I think what makes the difference is that Vanguards just have so little downtime. Charge replenishes your shields so quickly that, if you're not screwing up, you almost never have to stop to let yourself do a full recharge, when you're attacking, non-Claymore shotguns have point-blank power on the level of the Mantis/Widow snipes, without needing to reload after every shot or wait for enemies to get out of cover, and between shots you're bashing away with a surprisingly powerful melee attack. It's just a huge, steady stream of damage - of course, if you miss a shot or the enemy gets out of melee range or you take enough damage to break the Charge chain, you're fucked, but when you manage to keep everything going, you're a hot knife through butter.
I'm definitely finding it neat how there's a decent bit of a skill curve to the game, though. The first couple weeks, you mostly just heard "Vanguard sucks" from all the people who tried it as their first choice. Now that people have had a while to get used to the game and its systems, it's neat how the class balance holds up a lot better than anybody's initial impression of it.