Discrepancies between Kishimoto and the Asura Path?
Since your question has two parts, I will do my best to explain them.

(1)


  It turns out when Pain assaulted the Hidden Leaf Village, it pretty much was the end! So why did he say these things would not be in the Naruto universe when in fact they were?


Manga and Anime industry are like comics in western culture. Unlike those from DC, Marvel and other publishers, most of Manga artist (Mangaka) decides how the show is going on. I assume on Marvel or DC, they would need to discuss a lot with plotline for comics. 

Also you may have heard about Naruto pilot Manga, which is quite different from current implementation of series. What I am going to say is the "Mangaka" is king of the manga and whatever he says is not planned and will do anything as he desire in his manga, without considering things such as plot holes and others.

Plus with "Boruto" and Ootuski clan being from outer space will make these into making things like it is planned from beginning of series.

(2) 


  Also, what exactly are the powers for people who've wielded the Rinnegan at some point, like Obito, Madara, and Sasuke, regarding the Asura Path? Can they even use it since it involves constructing your own weapons? Or is there something they do to summon these weapons?


Well for this one, I have some scene from a book I read earlier. A guy asked "DO you know how strong is batman?" and his friend replied "As far as the author wants to". The same could applied here. I rarely doubt Kishimoto would really planned Obito to imitate as Madara at first place. He may have planned for him to be part of an arc, but not this deep.

Back to question, the only Rinnegan user who extensively use Asura path is, for what I believe,Uzumaki Nagato. Other Rinnegan users are listed but I have never seen any of them using such. 

For constructing own weapons, I believe is part of Yin and Yang combined release, not from abilities of Asura path. With ongoing "Boruto" series, your questions will be answered in more detail, so just keep patience. (ง ͠° ͟ل͜ ͡°)ง