Since celibacy was founded about 400 (granted it wasent codified till the Gregorian Reforms, but it was pretty much dominant since Leo the Great) while papal supremacy dident really become dominate till about 700?, if a pope (were they called pope that early) tried that, they would be completely discredited as a big leader, Byzantine would wipe the idea away fast in all of the areas they controlled, they had no use for it and it most likely would be a problem for them in the future, but I'm not sure that it would be so dismissed in Western areas.
I do think the idea is being dismissed a little too fast, if you are looking at it as a medieval all powerful pope having a dynasty generation after generation then yes, the idea would never work. However, the first step, getting rid of priestly celibacy, would make such a figure very unlikely to come about. I personally think the celibacy was one of the main reasons that the Catholic Church got as powerful as it was, if you dont have to support tens of thousands of family members for hundreds of years, you are going to be able to consolidate a lot of wealth and power. If for some reason the idea of priestly celibacy never won out, which I think is very possible, (1 Cor 9:5 or 1 Timothy 3:2-4 both support priestly marriage, and in the 2nd and 3rd century, priestly marriage was pretty dominant.) then I think this idea isent ludicrous. Marriage would localize power,(you have a family and you're more integrated into the community, marriages will make someone more protecting of local power, plus you need wealth to insure your children have good lives) and if a Vatican came about, I think it would be less powerful then it was OTL. I could see on the local level a lot of dynasty, a father passes the priesthood down to his son and that family is the local priest for the town. I could also see it more along the lines of hereditary for bishops, with them getting more independence and less central control, like the Eastern Orthodox. But as a OTL powerful hereditary pope, no, I cant see that.