I've never had that happen, even pulling a 6000 lb trailer. In fact it has a "trailer" button which provides more transmission-braking.
I did and fortunately I had the oil changed right shortly after with no damage done.........lesson learned.
I remember the old Desi@Lucy movie called The Long, Long Trailer. They were pulling a giant Airstream through the Sierra Nevadas with a convertible as I recall, and I think I remember a scene when Desi had his foot to the floor as they crawled all the way up a long hill at a walking pace. I wonder how many transmissions they burned out shooting that scene.
Your dad was talking about replacing worn brakes, not handling a downhill situation where the brakes have overheated and lost their effectiveness. B mode is there for situations that have a risk of overheated brakes. If you are still using a bit of throttle here and there, then you are far from that situation.
Even if there are no faulty parts, water vapor from the atmosphere slowly permeates through seals and hoses into the fluid.
One answer to 'why is there a B mode?' is that in the US, regulations require a gear position with more resistance than D. B mode satisfied the regulators. While it can be used well by a driver, you never need to use B.
If you are extending a glide, then there is no imminent risk of overheating the brakes, so it is not the situation that 'B' is meant to address.
Recently i visited hill station, here one of the track is very steep and rough, on uphill it was ok but on downhill i engaged b mode but still speed was way fast due to steep, my question is how to control speed on such steep hills, i miss first gear of manual in these hills, please help