True. In fact, I'm hoping that Weapon X is a going concern in the present-day MCU as well, and something that possibly Mr. Sinister is tied up with.
And as I think I've established, Weapon X was pretty recent for Wolverine at the time he joined the X-men even in 616 canon.
If you consider Logan getting the adamantium to be his 'origin story' as Wolverine, it actually makes perfect sense for it to be something that happens in the MCU's present or relatively recent past. Much like how Peter had only recently become Spider-Man when he showed up in Civil War.