No.
If checking is after 4pm, then at 4 and one second I expect to be able to check in. It doesn't say "rooms could be ready anytime after 4pm", but that "check in can happen anytime after 4pm".
If your interpretation is correct, then every DVC room could be ready at 11:59pm. That's not how it works. It means that the expectation and the goal is to have rooms ready at 4pm, like hotel rooms should be ready at 3pm. And in both cases, an issue might cause a delay, but it should be an exception, not the rule.
It was very sneaky of Disney to add the "after" some year ago, because officially they can say they haven't changed anything, while being ambiguous enough that some people will accept any delay. But it's a trick, we should keep DVC up to the standard of having the vast majority of rooms ready by 4pm and not accept anything just because of a sneaky word added to the website.
And to be honest, I do think the vast majority of rooms are ready by 4pm, otherwise these forums would be in flame with people complaining.