When information propagates fast relative to the rate of change of external conditions, the dynamic model converges to the stable point which would be the solution of the static model - are the models really different in any important aspect?

Instability is indeed eliminated by use of sigmoid functions, but then the utility gained from happiness (of others) is bounded. Bounded utility functions solve many problems, the "repugnant conclusion" of the OP included, but some prominent LWers object to their use, pointing out scope insensitivity. (I have personally no problems with bounded utilities.)