Interesting arguments. And not all hollow, though I would have used typhus as an example. Just because an animal can be infected by a disease doesn't always mean it is a credible source for transmission to humans. Or that they serve as a reservoir for other animals to catch it. Perhaps Norway rats "can" carry hantaviruses, but the CDC page is quite specific about the rodents that pose a threat to humans carrying it; deer mice one strain, white footed mice another, a third wild "rice rats" and "cotton rats". Toxoplasmosis pretty much isn't spread by anything except felines, though, or eating an infected animal raw.
The emphasis seems to be using the vague association of whole groups of economically or otherwise pestiferous animals with specific diseases as an excuse to campaign against noninvolved species, rather than as an honest evaluation of the amount of risk that might be mitigated by removing the animals involved. Raccoon roundworm poses a threat of causing human brain damage, but the simple fact is it mostly doesn't happen, while rodents that catch it can be debilitated for easy catching by the raccoons, and presumably often are. The threat of rat lungworm infection is also very scary as a rationale for controlling giant African snails in Florida, while the cases seem to amount to one man in Australia who swallowed live slugs on a bet; few people go around licking up African snail slime trails. Even relatively tiny numbers of sparsely scattered wolves are attacked as a threat by people using the scary possibilities of carrying the common dog tapeworm, while it is present in the much more dense dog and sheep populations, clearly an excuse for people who don't like wolves, period. Even the article linked to uses a domesticated version of a Norway rat as an illustration, though it isn't actually blaming the disease on "rats". It implies a shared blame, at least the possibility, with little or nothing to go on.