No, but you can go on short-term (or long-term) disability at many companies. There's an unfortunate pooling of sick time/personal time/vacation time at many companies today. But, in the US, it used to be absolutely the norm to have separate PTO and sick days and you were only expected to use the sick days if you were sick. (And they typically didn't roll over.)

And, if you left for whatever reason, you were paid out your accrued PTO (which often had an accrual cap) but not sick time.

Not really. The point of calling them sick days is that there is an understanding that an employee will use them only if she is sick. The limit is there because the business needs someone doing the job and after a person is sick for long enough, the business will have to find a replacement.