Well, the statement may be said in AW. As we know, Belthasar is a sneaky little Guru, since his Library shows up in the tower as a sort of third dimension to this all. However, if you could get straight to the TD from AW, then shouldn't the Time Egg be immediately useable without having to jump back over to HW?
Besides, Belthasar says a lot of cryptic things, and likes to play games. Remember his statement about the Time Crash? He says we might think it was a failure, but he "jokingly" asks us to think whether or not it may have actually been a success. His "finest hour"? He plays games...but you still have to trek to HW to use it. If AW could access the TD, then I would agree that AW is not in the DBT, and therefore part of normal space-time. But since you can't, and you have to return to HW, this means that AW is not in normal space-time. And the only place I know of off-hand that doesn't exist in space-time is the DBT.
I don't say a timeline in the DBT is "dead", just "defunct". Think about it. The Reptite Dimension was shunted to the DBT when Lavos fell, but it still remained alive; otherwise, Dinopolis couldn't have been brought over. Now I know that many people are going to say that statement is wrong, that the Reptite Dimension from which Dinopolis was brought over was from a different "base" dimension. But that in itself doesn't make sense. If the CT/CC base dimension was the Reptite Timeline, why would another dimension be the same base dimension? It makes more sense for the DBT timelines to still be active, but unreachable except through special means.
The compendium's encyclopedia entry for the DBT doesn't actually say that timelines sent there are "dead", in fact it only says they are "nullified". That doesn't necessarily mean it's all destroyed and no longer functional.
And also, the compendium's principle of discarded timelines entry states that although Marle's ghost said the ruined future is about to return, it was cryptic and probably only really meant that the destruction caused by Lavos was going to happen again, but not necessarily the exact same carnage. I don't think she could have actually meant that Lavos was going to erupt in the exact same way, and everything was going to go as it originally did.