I don't handle medical things well with my children. Once, when my daughter had to have blood taken from her arm. She was about 2 years old at the time. I had to hold her. They could not get a vein. She sat there just letting them poke her, however, after a couple of minutes, I said they had to stop because I was going to faint. I was able to lay down before I actually fainted but it was not fun.
My husband is very supportive and I have told him that he will have to help me with some of these things. He's ok with that. We have two kids.
If your husband is supportive and doesn't mind being the person in charge of illnesses, then I personally wouldn't decide whether or not to have another child based only on this situation. Of course, it is up to you guys.
I don't have a problem with cleaning up illnesses so I do that. But when it comes to certain medical situations, my husband takes care of that for me. As a matter of fact, I just told him that the next time our son goes in for a filling (he has one more, already did 2) that he will have to do it because I almost fainted last time.