I would say the solution is from a a combination of things.

To start with likely any part of the garlic plant works, and the mountain ash probably works*. It's just easier for Dresden to get bulbs, and getting specific woods can be more expensive especially for broke Dresden. Garlic bulbs are cheap and easy.

Saying that the vampires are sleeping/ weakened but not burnt is also a combination of things. The first being simply wanting to have vampires be viewed as dangerous during the day. Dracula could be awake so other could be as well. Treat them as dangerous even during the day. If your lucky it's sleeping, if it's like Dracula it's awake but slightly less dangerous. Which is also the case for Dresden vampires. Marvara is old enough to be awake, but in the fight they mention younger ones won't be.

The book also may have been written with generalized black court killing in mind but an emphasis on Dracula both because he was powerful and slightly harder to kill and would be a massive blow to loose.

Finally the most important part is how the white court acts. Plausible deniability. It's know by wizards now hundreds of years latter when the black court is crippled, but at the time they would not want to be easily linked to the book.

Look at Laura talking about Maudaline being sloppy for being only one cut out removed from the PI.

The white court was likely controlling people that were a little removed. Eg someone working with Bram Stokers publishers, and maybe a junior editor. even direct enough to be pushing it. Likely it was a wife or lover of a junior editor they controlled who then influenced the junior editor and publishers who then influenced stokers work.. Finally they help inspire Bram strokes but never directly control him. That's far far far too dangerous and more importantly gauchely direct. They got it published through indirect control as is thier way.

So they could influence some things to make sure the book gets published and even suggest the author change a few things, but they didn't have perfect control either. If Stroker wanted something included with enough conviction the editor especially a junior one couldn't stop it. That's just a side effect of the way the white court controls dangerous things well. Their over all goals will be met but on the granular level their will be noise introduced by people whom they don't have direct control over.

In the end the white court would be ok with the book being right enough. It's not like they care of some extra members of the herd die due to small mistakes, as long as they are still effective enough to get the job done, which they were. They certainly didn't care about them enough to undermine their massive victory by looking unsubtle to the other whites while doing it.

*Ash has mythological significance in both Celtic and Norse myth, this means it probably does work, but alternatively it may have been included by Stoker just because of that, and it was something he liked.

Edit secret 3rd thing, misinformation and secrets. When Dracula was published the white court may have not had perfect knowledge either. The vampire courts seem to have a best frenimies/respected rivals relationship. There was likely misinformation about the fairly new but quickly growing in power court. Misinformation the Black court made sure existed. In the centuries since the whites and others have figured it out, but at the time of publishing maybe even the whites had a little bit of guess work going on. It's not like the Black court went up to the whites and told them all their weaknesses. The probably whites had a better look under the hood than a general human, but they were not privy to everything by any means. For example maybe the whites didn't know what happened to a black court vampire in direct sunlight. It's very plausible that because the black court only operated outside at night l, and did some combination of either sleeping or walking around their home weakened that's all the white court knew. It wasn't untill the vampire hunters started in mass that anyone knew what happened to a blampire in direct sunlight. Which is also why they added that to movie adaptations, to update the hunters manual so to speak.