I can't pretend to understand all the ins-and-outs of targeting the JAK/STAT pathway. JAK3 (unlike JAK1 and JAK2) is found almost exclusively in hematopoietic cells, and is involved in a number of immune response functions (like chemotaxis). So the concern with targeting JAK3 would likely be various types of infections.
There is a different bar here for different indications. So something that might be acceptable for an oncology indication might well not pass muster for say RA.
To get a sense how complicated the whole JAK/STAT complex is, this picture gives a flavor:
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