The argument is only concerned with the proteins in tuberculosis bacteria, so this kind of claim about all proteins in all disease-causing bacteria is too broad to be necessary.

Whether or not skin irritation is usually a symptom of actual tuberculosis infection is irrelevant, because the argument is only concerned with reactions to the tuberculosis test.

In other words, there is no other bacterium which would lead to skin irritation when someone is injected with tuberculosis proteins. This is necessary because otherwise, we couldn’t conclude that a reaction meant tuberculosis and not some other bacterium.

Whether or not it’s possible to get tuberculosis from being injected with tuberculosis proteins is irrelevant to whether a skin reaction truly indicates tuberculosis.

What exactly it means to be “infected” by bacteria such that your immune system will recognize those bacteria isn’t relevant, because the argument doesn’t depend on that exact of a definition of “infected.”