Well, firstly, I think that He doesn’t require an invitation, only that choosing to let Him in is less painful than denying Him.
Letting Him in is like giving Him the admin permissions to your soul – He can mess around in there all He likes, reprogramming you to serve His desires. Or taking it away completely.
Yes, my version of Dark feeds on souls because He’s recuperating from His battle wounds, as He was dealt some serious damage and He desperately needs to absorb the energy they provide to heal. Being truly immortal He cannot die like we do… but He’s also silently suffering, in very real agony from the kind of wounds that should be fatal (and definitely would be to the likes of us).
When He’s hungry (and going on a rampage like in “The Shadow Over the Sun”), your immediate surrender is less painful and increases your chances of survival… well, as a demon, anyway. His forcibly taking it will cause a shock to your system that will definitely kill you. When He’s hungry, He becomes too single-minded to care about the consequences of His actions.
He doesn’t particularly enjoy the hunger, and tries to stave it off as long as He can because He doesn’t like drawing attention to Himself in this way. But He’s wounded and starving and, just like anyone else, is likely to snap in that state.
It’s a necessary evil for Him to take souls; He doesn’t want to, really, but He has to. He also doesn’t like causing His prey to suffer needlessly, so He’ll try to strike a deal with you so you’ll voluntarily let Him take it… making the process relatively painless.
And convincing more than 18 million people to let Him in means He’ll never go hungry again.