The BE-4 has been “sold” to customers but started development before Raptor, and is far behind it in technology and accomplishments. We won’t see a BE-4 flight before next year, if even that soon.
Oh, that's not true. They've filed a lot of litigation in that timeline!
Accomplished squat? They're building rockets and engines, mostly out of the public eye, with private money. How is that nothing? Because it isn't the exact path SpaceX is taking?
BO wants to land on the moon by 2024. Let's see what happens.
Something like the moon is a very iterative process. You need orbital flight capabilities, second and perhaps third stage capabilities, life support, launch abort systems, landing and take off from the moon, EVA gear and provable safe earth re-entry... just to name the few key ones off the top of my head. All are huge challenges and you have to solve them step by step with multiple test at each level.
The year 2024 is 4 years away. At BO's current test rate of 2-3 flights a year, and with a rocket that hasn't come out of the factory yet I can't possibly see how it could happen. I look forward to what they will accomplish but they have a very very long way to go.
Edit: adding that BO did recently win a contract to develop a lander only for the Moon and this is much more feasable. But it would still be docking with NASA's Orion for the actual trip there and back.
While I agree with the sentiment of your comment, BO is also developing with public money. It's true that their funding has largely been private, it's not exclusively so however.
"Blue Origin will receive up to $500 million from the United States Air Force over the period 2019 - 2024 if they are a finalist in the Launch Services Agreement competition, of which they have received at least $181 million so far."
"Blue Origin has also completed work for NASA on several small development contracts, receiving total funding of US$25.7 million by 2013"