Non power bounded generators of strongly continuous semigroups

It is folklore that a power bounded operator on a sequentially complete locally convex space generates a uniformly continuous $C_0$-semigroup which is given by the corresponding power series representation. Recently, Doma\'nski asked if in this result the assumption of being power bounded can be relaxed. We employ conditions introduced by \.{Z}elazko to give a weaker but still sufficient condition for generation and apply our results to operators on classical function and sequence spaces.