Yes, it does matter [when Paul lived], because people that knew Jesus were still alive, and therefore would have attempted to correct him or refute him, if his info about Jesus was incorrect.
El Cid
"Paul was creating a cult based on Jesus Christ, not Jesus. The only relationship of Jesus of Nazareth to Jesus Christ was the hijacking of his name and charisma. The three Synoptic Gospels which were written after Paul, are a systematic and largely successful exercise in correcting and refuting Paul.
It is impossible to read Paul with the synoptics open in three other bibles and find anything in the synoptics that supports anything in Paul. It is almost a trivial exercise in reading comprehension to find a refutation of Paul's misanthropy and misogyny in any random passage from the words of Jesus in one of the three gospels. If you do it in the other direction that is read the synoptics and try to find anything at all in Paul's misanthropy and misogyny that relates to the Jesus you find in the synoptics, you will find absolutely nothing.
I find the evidence for the existence of Jesus, the itinerant preacher and entertainer, persuasive. He would be a great televangelist today and as then he would refute all the Pauline garbage preached by the followers of his competitor in the religious leader industry."
What would Jesus do? If he were alive today would he have a television ministry based in a megachurch in Marin County? It sure wouldn't be in LaLa Land. Would he be regaling against the preachers of hate for your neighbors of the wrong religion, color, or sexual preference? Would he be successful?
I think the answers to all of the above would be yes. We are seeing a return to the gospels, particularly the Two Great Commandments in many local congregations in many of the big denominations. Certainly the bigots are the loudest and sell the lead in and follow up ads and so are supported by the networks or at least the cable companies. Someone foaming at the mouth at a gay person's funeral will get a spot on the news just like a train wreck. All the news companies care about is eyeballs, they don't care if the eyeball is blurry from booze or not, well, they do boozers buy. Train wrecks sell ads, and any train wreck will do. An emotional train wreck is as good, or maybe better than a steel one. They can milk it longer.