Gaim's part would have been so much better if the writers had done the most logical thing: Make an actual samurai movie with the Gaim Riders.
I can not understand why they felt the need to create modern sengoku era with machine guns, trucks and weird personalities.
As far as weird personalities go, really, most products featuring Sengoku era characters for younger or even teenager audiences seem to do that. Look at the Samurai Warriors and Sengoku Basara games (and anime in the later case), which often have rather "colorful" interpretations of the historical figures. Samurai Warriors has more classical interpretations for some characters (like Ieyasu himself), but often they can be rather out there too.
For the second one, I guess it's part of the way Toei has avoided going with completely alternate universe movies for a while. So, they couldn't just have the movie feature the Gaim cast from some alternate Samurai world and had to throw the tv show characters there somehow.
In spite of the eccentricities given to them though, both Nobunaga and Ieyasu here had clear references at least to the way they're often portrayed in pop culture (when they aren't portrayed as villains, at least), like Ieyasu's speech about unifying the country for the people or his answer to carrying on after losses. Both are very similar to things said by Samurai Warriors Ieyasu, and Nobunaga here even seemed to try to do the slow voice thing that Nobunaga has in both Samurai Warriors and Sengoku Basara.
I think something that definitely hurt the story though was Bujin Gaim. He neither mirrors any Sengoku era character, nor Kouta himself. He's basically just a random monster that looks like a Rider and that brings down the movie's entire conflict considering how he's the main villain. His presence also means that the general story here didn't really follow the flow of the Sengoku Era battles, in spite of some references (like Nobunaga dying at Honnoji, but without either being about to unify the country or Mitsuhide betraying him).
