Vanguards can solo a brute, it comes down to timing your attacks. When you see the Brute wind up to take a swing or charge, that's your cue to biotic charge. You'll be immune to their attack, and then you just roll away in the appropriate direction. Unless you start your charge too late, the brute's recovery time will be longer than your time needed to roll away. Shoot at the brute in between charges, and always pay attention to your surroundings and watch out for other enemies trying to interrupt your brute beat-down. The biggest danger is a ravager or marauder joining the fight.
I think the half-barrier nova is pretty much a necessity. Losing your complete barrier is a bad idea, UNLESS you start with a Nova and then biotic charge immediately afterwards. Or biotic charge, follow up with weapon fire for a second or two, then nova, then immediately biotic charge (since the cooldown is complete); as you mentioned.
Also picking the correct target to biotic charge is super important. It's not always about what you're hitting; often it's more about where you end up after the charge. It's usually better to aim for someone who is in the back of the pack, or off to the side, especially if they're near cover. Ideally the enemies are already shooting at your teammates, so many of them will still be shooting away from you after the charge. If not, duck into cover, wait for the cooldown, and charge at someone else that takes you out of harm's way.
Also if your teammates are shooting at a Banshee, and there's a Ravager and Marauder a short distance away... target the Ravager or Marauder. Your teammates are probably so focused on the Banshee that they'll not notice they're exposed to enemy fire. Even a bunch of Cannibals, Geth Troopers, and Assault Troops can do serious damage if left unchecked. Knock out those ranged enemies, so your teammates are free to obliterate the heavies.