It sounds as if your wife is in the primary training pipeline. It is true that recruits/trainees have a harder time getting to see a physician. One reason is that they tend to have lots of minor problems and also tend to magnify them. So they will often be screened by a very junior/inexperienced person. In the Navy "recruit sick call" was this type of 'cattle call' screening. They had to process loads of people, and typically unless your vital signs met some thresh hold you were sent on your way with some OTC. But then most of what they saw was minor stuff or people who were used to staying home from school if they had so much as a sniffle. The process is normally quite different once out of the schools pipeline.
To be quite honest, the story presented here does not sound quite right. For starters, she has pneumonia? Is that a professional diagnosis or self diagnosis? Pnuemonia is normally taken very seriously and usually (though not always) has clear "indicators", like a high temperature, that will get someone put in line to see an MD. Secondly, she *turned down* light duty? That will make eyeballs roll right there. They offered her a chance to get off her knee, and she chose to refuse the help because she doesn't want to get rolled back? Then her knee must not hurt that bad, many will reason. The corpsman/medic screening her will probably figure that if she feels she can continue training, she can continue training -- and there is no point in wasting the MDs time sending her to him.
She can get out of the military, anyone can. But if she is not careful it may be with a Bad Conduct Discharge or an admin-sep for Personality Disorder or any of a number of other outcomes that could have bad repercussions. If her knee is *really* a problem then she should go in and if they offer her light duty she should take it. If the time on light duty doesn't fix her knee up then the next time they will likely refer her. She can ask for scans (MRI, X-Ray) but won't necessarily get them, at least at first. Indeed, if she pisses off the medical people and they do a scan and it's negative, that can be used against her.
Talking about "refusing training" and wanting to get out of the corps will just make everyone more convinced she is malingering. Is she just looking for a medical excuse to get out? Be advised, trying that can be harder than it looks.