Finally I gave up and just started putting the car in neutral and breaking. This feels more natural to me and so it's what I do.

I'm sorry, but that it is a very silly idea. You shouldn't take the car out of gear when you are braking.

If you do this and something goes wrong you won't have proper control over the car immediately. You are going down a hill, you need to brake so you brake and take the car out of gear. Now imagine your foot slips off the brake pedal, or your brakes fail. You will now be left with a car that is accelerating down hill, until you reapply the brakes or get the car into gear again.

Also, think about what would happen if you are braking and then the situation changes and you need to accelerate again quickly to avoid a collision. Again you won't be able to do it until get the car into gear.

Finally, imagine you are braking and then someone steps out in front of you, or a car pulls out. You need to change direction to avoid them because you know you can't stop in time. To do this safely you are going to have to release the brake (unless you have good ABS maybe), again you will end up accelerating (or at best not slowing at all) towards the obstacle.

Taking the car out of gear (except to change gear obviously) while you are driving is not a good plan.

I personally use a combination of engine braking and brakes. On my MX5/Miata the front brake pads and disks lasted about 80,000 miles, the clutch made it to 90,000 (it wasn't the friction surfaces that went) and the rear brake pads and disks are still going strong at 130,000 so this can't be too bad a combination...