I wish I had held onto my copy of Systematic Theology to get more context.
That said, I’m not even entirely sure where he’s coming from on saints. Saints are those who the Church has declared have certainly made it to heaven. By being in heaven, they are obviously closer to Christ than we are. Sure, we believe that they were very close to God here on Earth, but that probably came less from some greater outpouring of the Holy Spirit and more them being more receptive to it. So at least in that regard, he appears completely off.
It would take more context to understand what he’s saying about priests. Perhaps he’s referring to ordination or the fact that the priestly life is often considered more capable of orienting oneself to holiness than married life, but I’m not sure that that would lead to being more “spirit-filled” as Protestants understand it.
With all of that said, do bear in mind that Systematic Theology is less a definite guide to Biblical teachings and more an exercise in what Grudem believes and doesn’t understand. It’s perhaps one of the more complete looks at the totality of belief to come from a Protestant in modern times, but Protestantism being what it is means that’s it has plenty of detractors even from Protestantism. I know many Reformed Christians consider him a heretic for some stuff he said about the Trinity and the relationship between men and women.* I remember getting really infuriated by his gross misunderstandings of those who accept the theory of evolution. Basically, Grudem may understand a lot about what Calvinist-leaning, conservative Baptists believe, but from what I remember, he has very little understanding of what anyone else believes.
* Just to give a very brief overview of this controversy, some Reformed Christians have adopted a form of subordinationism, in that they say that women being subordinate to men is a reflection of Christ being subordinate to the Father. IIRC, Grudem supported it in the book. There’s obviously a lot to the whole debate, but in general, many Reformed Christians think Grudem and others have adopted the ancient heresy of subordinationism.