The "brake before entering turn then on the throttle in the corner" can be tricky if not done properly. Adding power in the turn, especially after the apex, can take weight off of the front tire and cause a front wash out if care isn't taken. You see it happen all the time in televised race events.
And not using your rear brake at all, you're missing out on at least 10% of your braking power. Think about that ... without that extra 10% (or more) means you have to begin braking earlier.
I think some of these "schools" espouse that notion because they don't want new riders over braking with the rear end and creating a problem (over correcting when the rear wheel steps out a little bit, causing a high side). The rear brake is your friend. As I said already it adds at least 10% more stopping force. For example, if you're able to stop within (let's say) 100 feet from 60mph without the rear brake, you're stopping distance can go down to 90 feet from 60 mph with the added 10% of braking force. Those are hypothetical numbers to make a point.
Apply that idea to cornering and you see the point. I grew up on dirt bikes, motocross, and flat track, using the rear brake with front brake is natural to me (street, dirt, everwhat, no matter) and I use both brakes all the time. But if you're unfamiliar with the practise it takes time to learn. And as I mentioned previously, learn in increments. Do nothing abruptly, that goes for applying throttle mid-corner as much as it goes for pre-apex mid-corner braking.
I'm one of those nutjobs that prefers my bikes to have no ABS, no traction control, no slipper clutch. Take that into consideration when taking my advice.
EDIT: The idea of "I never brake in the middle of a turn" can get you killed if you're not willing to practice it. So you come off of a freeway and head down the off ramp. The off ramp kinks into a tight right hand turn with a "25mph" sign just before it. It's a blind turn. As you round the corner traffic is backed up to the point that there is a semi truck at a dead stop just out of sight around the blind corner. You're still leaned over, attempting to stop as the semi truck takes you by surprise. You'll be FORCED to brake in the corner ... or hit the rear of the stopped semi truck. If you do not practice this technique you'll not know how to get the bike hauled down before hitting the rear of the truck, or end up going down because your panic took over.
I watched a man die before my eyes in 1991 because he did not know how to use the brakes in a turn. He ended up in three separate pieces when the situation I described above actually happened. His torso split open as if he was in the Alien movie. Firestone exit of off the southbound 710 freeway in Los Angeles.