Apart from the obvious fact that the PSP is way too expensive/fragile/complicated to dominate the handheld market - I as a PSP owner believe it is doomed.
Why? Because primarily sony see it as a media device and a gaming device second, and you know what has killed the PSP - not the DS (although it sells more) but the ipod video...
I don't think PSP can dominate the handheld market because it simply isn't small enough. There's an important niche, for portable gaming systems that are not only portable, but small and convenient enough to be really practical to bring along. GBA is pretty much the only platform that can do that right now, apart from cell phones. GBA is a very limited platform, of course, and the limitations show particularly in the poorer third-party titles on the platform.
That said, for those times when that level of convenience is not needed, PSP is a fantastic gaming system - and with the size and quality of the screen it's easy to see why they're pushing it for media, too. Portable video players are mostly untapped territory, apart from things like portable DVD players (which are much larger than a PSP). They've just got to get more games for the thing. That'll happen in time - Sony's put enough into the system that they're not going to let it fail. The first year of a console's life the titles are always a little sparse...
Personally, for games, I'm more interested in the DS at this point. The PSP games I might have been interested have all turned out to be something other than what I was hoping for (like AC: Formula Front and Metal Gear Acid). There are still some impressive titles I'd love to have, like Mercury - but for the most part the DS's unique capabilities are a greater draw for me. Likewise for homebrew: the idea of storing homebrew on memory sticks is attractive, but I don't want to deal with a homebrew-friendly PSP getting re-flashed when I run a new game... With the DS that can't happen. Software can't re-flash the firmware unless a jumper in the unit is shorted. I don't know of anything already home-brewed for the PSP that I couldn't live without on DS (I'm guessing the arcade emulations for PSP are a lot better, though) and for my own home-brew work, I'm more interested in DS's novel capabilities than PSP's raw power. But I want to stress that that's just
me it doesn't reflect on the vitality of the PSP homebrew scene as a rule.
But I really believe the best for the PSP is yet to come. It's absurd to declare the platform's life is coming to an end. It's just starting.
As for the comparison with the new iPod, I think there is no comparison. It's not even worth making the comparison. PSP would be good in place of a portable DVD player, for long train trips and odd times at conventions... iPod is much more convenient to carry and use, but the tradeoff is there with the smaller screen. It plays video, but it's still a music player at its core.