MC44603 is crap. Its built-in transistor can't switch mains voltage and it lacks several features to conveniently and safely drive external MOSFETs: UVLO, push-pull output, primary peak current limiting.
Popular oldschool options are:
Viper series from ST: built-in MOSFET, few external components, good enough for a few watts
UC3842~3845 series: very common, needs external FET
NCP1207, NCP1377 are another controllers I recently found in off-the-shelf PSUs which appear to be fairly available and have a nice bonus: secondary overvoltage detection and shutdown.
There is a lot of newer chips too, mostly from China, but they are more obscure, sometimes without good datasheets, often only in SOT23-6 which you may or may not like.
Whatever you do, beware of capacitors charged to 330V and remember that it's difficult/expensive to safely probe primaries of mains-powered PSUs. Such SMPS is not an ideal beginner project. Honestly, I probably wouldn't know how to design one properly, I have only ever repaired them.