That example sentence is fine. Compare to the instruction from the Bible in John 8:7, “He that is without sin among you, let him first cast a stone at her.”
But that biblical quote is from the King James translation, which is three hundred years old. To modern English speakers, your example would be more naturally expressed as He who is always shooting must sometimes hit. And the use of the masculine to mean anyone of any gender is far, far less a convention than it was even one century ago. So yes, One who is always shooting… might be better still.
But… he who does remain common in at least a few aphorisms. I have not yet encountered any reworking of He who hesitates is lost.
As @JamesK writes, your example is in a fairly formal style that would not be commonly used in everyday speech… unless, for example, it’s in an aphorism. That’s a context in which that kind of formal (old-fashioned) style can be appropriate because the style itself implicitly helps communicate that the utterance is indeed an aphorism.