I've had several stylist sims and they have reached level 10. The first one I did without cheating and it just took me awhile.
Perhaps even though the action is not in the cue, you can try to still go to the house of the sim who wants a makeover and click on the sim themselves to further prevent the action being repeatedly ignored and canceled. That way your stylist sim is there at the house and has nothing else better to do than their job!