The Hashiras would likely show concern. Some, like Shinobu, would immediately start looking into possible medical solutions. Giyuu might be more reserved in his worry but would still keep a close eye on Tanjiro. The other demon slayers would also be affected. They look up to Tanjiro, so his sickness would lower the morale of the group. They would try to support each other while waiting for Tanjiro to recover.
One main element is his difficult upbringing in poverty - stricken surroundings. Another is the constant bullying and discrimination he endured due to his physical appearance. And of course, his strong bond with his sister Daki. This bond was what drove him to protect her at all costs and also influenced his actions as a demon. For example, he would go to great lengths to avenge any harm done to her because of their shared experiences of being outcasts in society.
Inosuke doesn't become a Hashira in the manga. The story progresses with him having other significant experiences and achievements instead of achieving that specific position.