Bodkin arrows are able to pierce their armor, though it is still a good idea to bring an armor penetrating weapon for them. The reason early game archery is hard to use is because the character is likely stuck with crafting low quality bows that are thought of from intuition rather than books. When you get Bowyer’s Buddy, you can get the recurve bow quite early, which deals with zombies quite well. Once you get Fletcher’s friend, you can create the metal arrows, too. Broadhead arrows are good for unarmored targets, bodkin arrows are good for armored targets, and once you get an extremely powerful bow, target arrows become more useful due to the increased range and frequency of critical hits thanks to their accuracy.
Also, there seem to be two kinds of bows, the faster ones with low damage, and the slower ones with high damage. You might be taking more shots to kill zombies because you are using the faster bows, which are probably better to use against enemies that have guns, since you can shoot first. The slower bows often deal more damage.