Question: Why isn’t it a problem to put tzitzis onto a garment which has shaatnez and then take out the shaatnez?
A. The problem is that there is a rule that one can only put tzitzis on a garment which already requires tzitzis. For example if a garment has rounded corners which are not obligated in tzitzis and he tied tzitzis on each round corner and then cut the corner so it would be square which now makes the garment obligated in tzitzis, he must now untie the tzitzis and retie them to be allowed to wear the garment.
B. The main problem is that you can’t perform a mitzvah through doing a sin.
C. The prohibition of meat and milk has to do with a mixture this is the same prohibition of shaatnez, a mixture of wool and linen.
Answer: The issue of shaatnez and tzitzis do not cross paths for the prohibition of shaatnez is in the mixture whereas the mitzvah of tzitzis could be performed within the measurement of wool by itself, assuming most of the garment is wool and there are just a few strands of linen mixed in.