Since there's no actual canon reasons, all we can do is homebrew solutions to them.
Ah, I didn't realize you had put agreed with the sentiment that there is no real purpose to the Evil Gods, beyond having them to have them. Yes, if you have reached that point then the only thing left is to homebrew solutions, and I liked some of yours in theory. I had my own that I implemented.
No, because their war involves supremacy between goblins and orcs, both of which are Material Plane creatures. Even if a lot of their fighting takes place on the outer planes in Acheron or whatever, the basis is the Material.
Just checking, because the end of the Blood War is generally considered to be the point when the winning forces turn to the Material plane, so I'm not seeing much of a difference here.
I'd say it's less "cancels out their ability to focus on other things like the Material Plane" and more "they have no need to be interested in other things."
Take this as a given: gods either need or want prayer. Whether it's food for them, the divine equivalent of Likes, or something else entirely. To get prayer, they want living mortals to be their worshipers.
But arch-fiends don't need or want prayer. What they want is souls, which are food, a power source, and a currency all in one. Fiends don't need worshipers to get souls, because evil mortals who die go to the lower planes anyway, no matter who they worshiped in life. Now sure, a person worships an evil god, they're probably going to instead show up as a petitioner in that god's realm (although in my mind, an evil god would probably not accept the souls of worshipers who were evil but who failed to actually follow the god's portfolio--Bane wouldn't want a worshiper who spent his life murdering puppies for fun but who shied away from war). But imagine some dude who worshiped Tyr but went too heavy-handed and his "justice" turned into torturous revenge. Tyr would likely kick him out to the lower planes. Unless he becomes undead, his soul goes to the lower planes and becomes a larva. It then gets rounded up by a night hag or something and stuck with a bunch of other larva in a cattle yard, and then, eventually, sold to whatever entity decided to buy it--which could be a devil, demon, yugoloth, or anything else that has a use for damned souls.
Arch-fiends also get worshipers. This is a bonus because it both gives them a mortal agent who can increase the amount of evil in the world (thus ensuring more souls), and when the worshiper dies, they're likely to then show up in the fiend's domain--which means they manifest as a devil or demon. Even a lemure or nuppirubo is more useful than a larva in the archfiend's fight against the other archfiends, and some worshipers will end up as Type II or III fiends instead. But, these worshipers are not essential in the way they are for gods. They're just icing. Arch-fiends get the cake automatically, where gods have to work to get it.
An interesting way to take it. Except that the Devils do make deals for souls. Sure, maybe any given Lawful Evil soul gets taken to the Nine Hells, but then it is just picked up by whoever. Devils who don't focus on the material plane to some extent tend to end up weaker, unless they have something to offer those hags, because they have to fight and search for souls, while those who make contracts are guaranteed souls.
I'd say the same thing happens with the Demon Lords. Worship them, pledge to them, and your soul is send directly to them, instead of them having to grub in the dirt for a larva. It isn't icing, it is delivery.
Also, while it may have fallen out of style, there is the divide of the mortal into soul and animus, which means that they might be able to get the soul for power, and use the animus to create a fiend, making it a two for one deal.
So, not essential in the same manner, if we go with the idea of a god without worshipers dying, but still important enough to be a priority to them as well.
Maybe, maybe not. I have no idea what the arch-fiends did before the Blood War was invented for Planescape. But let's say that the Blood War was never invented. No Lawful fiends vs. Chaotic fiends. The arch-things would likely still be focused on the outer planes instead of the Material. They would be fighting other arch-fiends, if only for territory, resources, or to settle grudges. They'd just be more likely to be fighting other fiends of their own type. There would be probably more attacks on or by celestials. With the Blood War, angels are content to ignore demons and devils as long as they're killing each other and not other beings; without the Blood War, they'd have more work on their hands.
So this, combined with their lack of need for living worshipers, means that arch-fiends still don't have any reason to hit the Material Plane.
I disagree entirely. The Material plane is the only source for more devils, you may have them showing up randomly in the Styx, but that is much less efficient then getting them to pledge directly to you, denying all your rivals the potential to obtain those mortals.
Demons might care less.... except that they are embodiments of destruction, they have an intense desire to destroy the material plane, by their very nature. So, there isn't really a reason for them not to go to the prime, at least as much as they currently do. After all, they don't just need resources to fight Demons vs Devils, but also Demons vs Demons and Devils vs Devils... conflicts that are still ongoing despite the Blood War