Can the SMPS IC works only with power from startup resistors without using Aux winding power?
Depends on the part and the design of the circuit. There's no general answer for "SMPS ICs" 

Many could work without an aux winding supply at the expense of size and efficiency. (The resistor would have to dissipate lots of power all the time.) 

If you add some specific details we can make the answer more specific as well.