How do you know what designers or driving instructors were thinking? Is this just what you think their intentions were?
Maybe portland_bill can give us a first hand instructors reason? An instructors handbrake gets a lot more use than the average car.
Personally I've had to change the handbrake on two cars I've owned, both shortly after purchase. One was a worn ratchet rack the other was a worn pawl.
Both parts were made of steel (probably mild steel but I didn't test it) which does wear. Comparing that to Cast iron or cast steel brake disks and friction pads is just nonsense.
When first in the motor trade, we used to replace handbrakes often enough for it not to be a rarity. Always due to wear to the ratchet, either the rack or the pawl. At that time it was always recommended to push the button, llift the lever, release the button. They were made of mild steel, with no added lubricant, just the oily film from the pressing.
I think it was the Peugeot 307 that started the trend for not pressing the button, but that was not a cure for that car. With rear disc brakes, if the handbrake operates on the pads, a significantly hot disc can release enough pressure as it cools to allow a car to move. Cars with separate handbrake shoes within the disc do not suffer from this. With a few 307s wandering away on their own, motor manufacurers panicked and changed their recommendations.
As I said before, I see many drivers, new and experienced, that release the lever a little as they release the button, often dropping the ratchet down one tooth. Not pressing the button alleviates that. I prefer to nag my learners incessantly until they adopt a proper technique, and also understand why. Those that struggle will find themselves parking on a lot of hills until they get it right consistently. Great to see a bit of panic as a learning tool.
Having changed a few handbrakes in the past, it is a horrible job. Working both inside and underneath a car makes it difficult to keep dirt out of the car, as well as being fiddly between the seats and above an exhaust. Additionally, as the handbrake is applied without pressing the button the noise is painful to my mechanically sympathetic ears. For these reasons I will resist not pressing the button.
With more emphasis on blaming others when a mistake is made, people look at vehicle manufacturers as culprits if their car runs away. The manufacturers have reacted to peoples' inability to keep pressure as the button is released by recommending the lift without pressing technique. Like so many of the 'driver aids', we are trying to mask poor technique, rather than educate to do it properly. The result is likely to be more people with poor technique.