In my opinion, (modern) zombies are difficult to get right. By modern zombies, I mean everything post-Romero... undead zombies that attack and consume the living. (Classic zombies, the singular enslaved person stolen from his grave by a warlock/sorcerer... well, best to not even attempt those unless you're willing to do your research).
The trend has been to try to scientifically explain zombies. They are still alive, though their biology has been altered. Maybe by a virus or other pathogen, or even a toxin (or a nanobot!). Even if you can do that and do it well, it robs the zombie of what makes it a compelling and disturbing concept: humanity's inability to do something about it or to understand why it is happening. Zombies are supposed to be inexplicable and incomprehensible. Inexorable.
That makes for difficult story-writing though. So what can you, as an author know about zombies? Well, they're not the only kind of undead. Of things that can definitely be called undead, we have at least two other varieties: the classic vampire, and the classic (animated) skeleton.
At least one of these is known to consume blood. So yeh, it could make sense for zombies to drink blood... but not for the reasons you provide. The undead consume the flesh of the living for reasons largely unknown beyond that they "hunger" for it. It's related (though perhaps not directly causal) to others also becoming undead. Maybe just because when they are gored, they tend to die... and in some mythologies, people who die (or those who die because of the undead) become undead themselves. Or maybe it's because the undead are contagious somehow... but not in the modern sense of the transmission of pathogens. A different, more primitive concept of contagion that almost resembles defilement.
Does a zombie need to pump blood? Why? If your zombies look anything like modern movie zombies, there is zero circulation going on. The sorts of decomp those exhibit is active/advanced, often even into "dry remains". No blood could flow in such a corpse. And if it did flow or need to flow, how could slurping it down into the stomach then send it out into arteries and veins? Nothing larger than molecules of fat passes through into the circulatory system... often, many important medicines must be directly injected because the digestive system destroys those.
I suggest that maybe you've confused two different tropes. Instead of the zombie trope, you want the "mindless drone" trope. It works to accomplish many of the same elements you are pursuing. You get that brainless hivemind vibe from it, where masses of (seemingly unthinking) people can form an inescapable mob. You get the horror of people imagining becoming part of it themselves, and losing their personalities in the process. But, if your main character needs to be a thinking person, he could "wake up" from it. And be the individual caught in that mob, trying to hide and hoping none of the other mindless drones notice.
Obviously these two tropes are similar, but the latter doesn't have all the baggage of hundreds of years of mythology and folk tales that the former insists upon and that everyone is familiar with.
