At MAX power, there's a lot of very hot air going into the system. At some point, it's hot enough to do some damage....perhaps softening the metal or doing some long term structural weakening. At idle, the temperatures are much lower.
But....in most cases the pilot is much too busy to use a "sliding scale" of TAT limits for it's (anti-ice) use depending on the power setting...so only one TAT "limit" goes in the manual. Therefore, a "one size fits all" limitation is published.
The TAT gauge is not calibrated like a Rolex, and may be off a few degrees. So 11C is not going to melt any systems, nor is 9C going to be perfectly OK for the life of the airplane. A little common sense has to come into play here. I've seen on occcasion (on descent and approach where the OAT and IAS vary continually) crews turn it off at 11C, on at 9C, back on at 11C, etc. This is overkill.
The very few conditions where ice can form at 14C are very short lived, and very little ice can acccumulate. This again is overkill.
I'd suggest respecting the published limitations.