There is no need to increase thrust if all that fails is just the one engine.
In the sanitized world of full flight simulators, this is true. The problem that I have with it is that real emergencies seldom are as cut and dry as the simulations. The sim really doesn't model a big piece of nacelle that might be hanging off due to engine damage. When it happens, you don't necessarily know "why" the airplane isn't performing, you just know that it needs more power. With a derate, you may just be in " no man's land".
The effect of slush / spray impingement during takeoff is a retarding force, thus greater takeoff distances will be required. However this might help in an RTO, but then again the conditions may be inexact, the braking action incorrectly assessed/reported, and more runway already used in the takeoff roll.
You are correct. I could have been more succinct about this. The retarding force is more of an issue during acceleration. It's going to take you longer to get to V1 so you'll have less distance to stop. The drag will help in stopping but effects of braking will still be reduced due to reduced coefficient of friction the exact value of which is difficult to determine, thus the desirability/requirement for a reduced V1 value. I was merely trying to explain how an increased payload is sometimes available for a derated takeoff on a contaminated runway.
Regarding slide 13, it's been a while since I've flown an Airbus and it was the older variety (310/300). Also, we didn't derate anyway so it wasn't an issue. I am unaware whether AB uses derates to improve payloads on contaminated runways but the original question seemed to imply that they do. The slide you referred to would seem to imply that they don't. Personally, if I were "King of the World", I would only reduce thrust on takeoff using the assumed temperature method as that would always make full thrust available if needed. For most airlines, it probably wouldn't matter much but if a lot of their operation were from contaminated runways, I could see the commercial value of using derated thrust.