I'm a Classicist (and non-Christian), hadn't heard of Richard Carrier before, it's an interesting list, though the guy's anti-Christian bias infuses it so much it's a little hard to take him seriously. He's got some odd conspiracy theories about Nero, for example. And for someone with his perspective you might imagine an intact copy of Josephus prior to the later Christian edits would seem like it would be #1 on the list, but since he's a Mythicist he doesn't mention Josephus at all, since in Josephus you have a few mentions of Jesus as a historic figure which is evidence against the Mythicist belief, esp. given the chronology.
Ancient Rome was such a hotbed of conspiracy theories that familiar tropes today, such as Nero's slaughter of Christians, stand on extremely shaky ground. The texts we have today were selectively preserved, and unreliably transcribed. We have exactly zero original manuscripts of any of what historians are obliged to rely on. Everyone writing the history preserved was biased, and much of it was doctored by medieval monks, often by accident. E.g., we now know, reliably, that a single-letter transcription error produced "peace on Earth and goodwill toward men" from the original (in Greek, of course) "peace on Earth for men whom God favors".
Every last scrap of documentary evidence of the beginnings of Christianity was deliberately, openly, proudly torched by the early Church, so every conventional fact stands on extremely shaky ground. Consensus among current scholars not pledged to uphold Church doctrine is that the entirety of the New Testament, absent about half of Paul's letters, is wholly fictional. Paul, of course, had his own biases. It is still possible that some anecdotes are based on factual events, but we have no reliable reason to believe any in particular. Opinions vary.
People invested in current consensus should want to have actually-reliable documentation for any factual basis for any of it. It is just possible that there really was a Jesus from Nazareth at the center of Christianity. It is even just barely possible that this Jesus was crucified, maybe even in something like the manner now believed. (Zombie dinner parties afterward, not so much.)
The guy's a crank-historian and if you stick with him you'll follow in his footsteps. The ancient world is certainly quite interesting, I can tell this guy snarks at Ehrman, it's worth reading Ehrman on his own since he's a professional historian rather than a crank, and it's with looking at what else is out there that professional historians are up to. There's the chance of doing good work outside the academy, but then there's this Richard Carrier crank kind of pop-history preaching to a specific kind of converts.
The one reliable thing you can say about Richard Carrier is that surprisingly many otherwise pretty-sensible, competent people, Ehrman among them, absolutely lose their f'in minds when anything about him comes up. He might be wrong about some things, but he is certainly right about others. Which each is will become clear as more books are restored from Herculaneum.
Carrier is unpopular among Christians and crypto-Christians for obvious reasons that should not affect the rest of us. He is a well-taught, peer-certified, and articulate historian who is unafraid to take up opinions unpopular with retired colleagues. There is no reason to lose your mind about him: if you disagree with him, identify specific reasons why, and express exactly those. Be prepared to deal with Bayesian statistics reasoning, something old historians often have difficulty with.
The Historicist vs Mythicist debate is not about Christian vs rationalists or whatever you imagine. It's about evidence-based history vs conspiracy theorists. Using Bayes to rationalize the conspiracy of an invention of Jesus is fine with him since he's a crank and no professional historians are engaged with his work so he can make up whatever conspiracy theories he likes. In the end the Mythicist position always requires a conspiracy to explain why the documents we have exist as they do, while the Historicist position would apply Occam's Razor to inventing that conspiracy.
Anyway, the reality is that Josephus exists, a Jewish historian writing about the history of Judaea less than fifty years after Jesus's death who mentions Jesus's existence. The Mythicists have to deny what is actually slam-dunk evidence by ancient historical standards to hold their position which is why it's always been the territory of cranks.
It's really funny that in this day and age it's atheists who are driving along tedious religiously-motivated debates about the ontological nature of the person of Jesus. It's doubly funny that the self-proclaimed rationalists cling to a conspiracy theory and deny evidence. Anyway there's a lot more to the ancient world worth learning about, but I'd recommend learning from historians who are professionals to avoid openly biased cranks with a religious axe to grind.
We have no reliable documentary evidence. The Josephus we have is all based on a single manuscript held by Eusebius. The consensus position of historians is that the Testimonium Flavianum is, at minimum, heavily doctored, so is objectively useless as evidence one way or the other. Other independent sources are all similarly copies of copies performed by monks. Thus, any conclusion is no better than opinion. Nobody needs to invent conspiracy theories about the early Church: all the early sources we have absolutely revel in conspiracy theories, and boast of torching whole libraries of heresy.
Mainstream historians, as a rule, don't touch Christian origins with a ten-foot pole because the independent evidence remaining, a tiny handful of quotes that all fit on literally one page, stinks to high heaven, and is guarded by legions of hornets' nests. They have plenty of more productive ways to spend their time. Does it matter, ultimately, if a Yeshua existed at the center of the surviving cult and was executed, the only factual detail of any significance?
It is, as they say, moot. Only actual, reliable evidence could ever change that. We used to think none could ever surface. Herculaneum could very possibly change that, and much else of overwhelmingly greater interest.
Carrier has very interesting things to say about the non-biblically-adjacent finds likely to surface there. However distasteful you find his biblical opinions, he is among the best choices to read on topics of late Roman science.
As a little background, I did an MA in Classics (focusing on ancient philosophy), and started a PhD. I'm not a Christian, and mostly ignostic - religious debates are a waste of time. Since then I've kept reading, I have a lot of Ehrman's titles, and I've read a lot of the sources in Greek. While I started with more of a focus on philosophy and poetry I've gotten more interested in history and have kept reading for a few decades since.
>Carrier is unpopular among Christians and crypto-Christians
Carrier is in the minority because he holds a minority position. The reason it's a minority position is because it's steeped in conspiracy theories even if he tries to use Bayes to add a fig leaf. When I was working with them, no historians I knew of back in the '90s were crypto-Christians pushing a Historicist agenda, some of them were indeed priests, but no one was hiding anything. I am very sure you can't point to an actual working historian who is a crypto-Christian, while that term introduces even more conspiracy thinking.
The Historicist agenda isn't actually some secret formalization of traditional Christian doctrine, it's not a Christian or a crypto-Christian position, it's a position useful for historians speaking about secular history, that's why it was developed, and why it's the consensus position. It's an evidence-based position with sufficient evidence that professional historians hold a general consensus and the only outliers are folks who are motivated by a kind of evangelical atheism where they are the ones actually dragging religious biases into the debate and rejecting evidence-based thinking in favor of conspiracy theories (flavored with Bayes).
"Crypto-christian" just means saying they are not Christian, yet still assuming things just because Christians insist upon them. Bart Ehrman is a good example. You can shriek "crank" all you like, but either you engage with the evidence, or you are a crank yourself. Carrier exposes all his evidence and all his reasoning for public inspection. Ehrman just huffs and blusters disgracefully.
Your assessment of a "minority position" carries exactly zero weight. Either the gradient is positive, negative, or static. For the case of mythicism, the gradient over the past century is strongly positive: a hundred years ago nobody could even discuss mythicism. Twenty years ago, few found it plausible. Since then, the number has risen sharply. As Max Planck said, science advances one funeral at a time. The trend is clear.
Does it actually matter? Only as much as any history matters.