No, that's not a problem. Pulmonary infections of yersinias pestis can take a much longer time to gestate than when it's introduced to the blood. You can travel, symptomless for a week, breathing on people and spreading it. That's it. There's no mystery here. People traveled much faster than the rats. For example, on the back of a horse, which would also be a carrier, as a mammal. The rats never needed to migrate to get it done. They didn't even need to arrive by ship once it jumped to humans. Because of how much time humans spend with each other, and other mammals.

All you need for an outbreak is a sudden decline in the rat population, which then results in the fleas jumping to other species at much greater number. Fleas don't really care what they bite. It's just that rats live like humans in swarms, piled on top of each other. So rat blood is just the most widely available, and widely accessible food source... until it isn't. So really, when they try to make a point about the climate suggesting the opposite of what we see in the historical record, they're shooting themselves in the foot while missing the elephant in the room. It's not strange at all that conditions that would stress the rat population would result in higher transmission to humans. It would also result in higher transmission to dogs, cats, sheep, pigs, donkeys, horses, cows, etc... You know. The list of all the animals humans spend the most time with.

It vectored towards us because we were what was available. And when it figured out how to infect our bronchial sacks, it really spread like wildfire. Think about it. Humans are built to spread pathogens by breath. Especially in a world where the literacy rate is in the single digits, and there is no mass communication.

News traveled by word of mouth.

Read that sentence again. That's the elephant in the room. If a town found out about a neighboring town being afflicted with plague, that necessarily meant that someone traveled in person to say it, and breathed on people to do it. That's it. There's no mystery.

edit: spelling