You mean our world has orbital nuclear platforms, genetic supermen, interplanetary spacecraft that look like submarines and the rest? Wow, who knew?
What Trek tends to do is ignore what they have shown us about the 20th century except for things like Khan coming from that time.
What are they going to do when we don't have sanctuary districts and Vulcan spacecraft landing in Montana? Or when we don't discover warp drive or dilithium?
Those would be the parts that are ignored when the real calender catches up with Star Treks continuity. Not always, but most of the time. You know, like I said in my second paragraph.
What about when we discover that you can't travel faster than light (you know, like now)? Or that a human/alien hybrid is just silly (again, like now)? Or that turning someone into energy in order to reassemble them someplace else would just create a big BOOM (once again, like now)?
Leave Trek be what it is. If we require something to replace it let's have something new, like Roddenberry did back in the 60's.