'who' is the correct relative pronoun. Shakespeare lived 500 years ago, and although his writing is absolutely beautiful, you shouldn't take it as a guide for 21st century English.

„He that“ doesn’t work because „he“ is a person and „that“ refers to objects, even though native speakers get it wrong all the time. „The people that said that are crazy“ instead of „The people who said that are crazy.“ A very common mistake. However, saying „He who“ is not technically wrong, it’s just a bit archiac sounding. „He who hesitates is lost“ is an idiom so the rules don’t really apply there. I always tell people English is a language of brevity. It hates long words, complicated constructions, and long sentences. It’s a matter of style rather than grammatical correctness. So let’s take your „He who“ expression. Not really grammatically wrong, but ask yourself „Is there another way to say this that’s shorter and easier?“ The answer is yes. How about „Whoever“ rather than „He who?“ One word, simpler, easier. Even though there are 7 letters as opposed to 5, you’re only using ONE word. That makes the English speaker‘s ear happier. Weird, but that’s kind of how it works. And that maybe why „He who“ sounds a little archaic, outside of the idiom you said, which you can still use.

John has just spotted out the problem.