Novels are not canon to the series. And they never mention how many swords Sparda had, DMC implies that he had Yamato too when we use his costume, but again, they never mentioned how much swords; at that time we only had clue to know Yamato, and as a matter of fact FE/Sparda.
Trish really mention that he "lost his mother and brother to evil 20 years ago", which is inconsistent but not really contradicts DMC3 as it's not even a clear statement about all their past; as the story remains the same that Mundus (ordered to) kill Eva and separated Vergil and Dante in some way that we don't really know.
But that should hit the point home: DMC3 never tried to wipeout what DMC left on its background. It worked what already existed and they've build a game based on what Kamiya left for them, thus narrating an story prior to the first DMC; that's why it's a prequel and not a reboot.
Also, i think you confused yourself, as Nelo Angelo is Vergil himself in his corrupted form, who defeated Vergil was Mundus.
Even if there's some incronguency between dialogues and events, it kept all the continuity of the series. DMC1 still happens after DMC3, that's why they leave an opening to lead to the exact events that happened in DMC1; they've never touched what happens on DMC1 and is still a part of the timeline. A reboot should have to make "another DMC1" to continue that story. I think that should hit the point home and show why DMC3 can't be considered a reboot at all, even if it changed some things (like that implied past of Vergil in Trish's dialogue, that Dante itself not confirms that was exactly what happened) that would be later clarified by Kamiya, if it's not shown in official media it's not part of the story at all; what Kamiya had in mind is not what he showed to players, if he envisioned something different it's completely another thing, the official Devil May Cry is what we see in the games and whatever Capcom decide to consider canon (like the recently confirmed information that Nero is indeed Vergil's Son in the japanese artbooks, although no one knows who is he's mother or when she get pregnant of him).