Is it true that modern, purebred chickens can't hatch their own eggs?

First, broodiness is not tied to whether a chicken is purebred or not. Broodiness depends more on a chicken's breed, her genetic background. For instance, Rhode Island Reds very seldom go broody. Leghorns and other Mediterranean breeds are very unlikely to go broody, too, because not going broody very often is just a characteristic of those breeds. Other breeds may go broody very frequently--too frequently, some people find! For instance, orpingtons and silkies often seem to be broody most of the time.

Second, as for "modern" breeds being less likely to brood, that's not true, either. Leghorns have been around a LONG time, and hardly ever go broody. Ameraucanas are very modern breed--approved by the ABA in 1980--and yet they will go broody and raise chicks. Broodiness doesn't have anything to do with how new or modern a breed is. It has to do with whether that breed was selected to retain its broodiness. Breeds selected specifically to be the best layers (like Leghorns, RIR, etc.) seldom go broody.

Differences due to breed happen with any pet, not just chickens. Dogs and cats have different characteristics based on their breed, too. You wouldn't expect a chihuahua, even a purebred show quality chihuahua, to be able to herd sheep very effectively. If you wanted a dog to herd sheep, you would instead get an Australian Shepherd or a similar breed with a herding drive and high intelligence. If your chihuahua doesn't herd sheep, that doesn't mean he's not purebred. It just means his genetic background doesn't include a strong herding drive.

So, in order to get chickens that will go broody, simply choose breeds that are known for broodiness. Cochins, Silkes and Orpingtons are among the broodiest of breeds, so those might be good choices for someone who wants very broody birds who will hatch fertile eggs. Do keep in mind that there are varying degrees of broodiness, though, so if you do want birds that will go broody very often then you will probably want one of the three breeds I mentioned above. Other breeds listed with broody qualities will not (usually) go broody as often as those. However, individuals can vary! So, just because Australian Shepherds are usually good working dogs doesn't mean that every one will be that way. It's just more likely... and it's less likely that your chihuahua will be a champion herder. There are no guarantees, though.

So, whether a chicken goes broody or not has nothing to do with when her breed was developed or whether she is purebred. The broodiest bird in my home flock this year is a mixed breed: half Silkie and half Faverolles!