This has been a common technique among some 737 operators for years.
If memory serves me right Boeing discussed this in a bulletin years ago. Obviously well hidden from many seasoned 737 pilots, I know of a lot of experienced 737 guys who don’t do this. I can dig out the bulletin if anyone needs the number. The essence was:
Recommended to have minimum time between packs off and engine start and between start valve close and next valve open. This to prevent large turbine wheel temp fluctuations which has lead to premature changes of 737-300/400/500 APUs due to rotor problems.
By the way “AIR CONDITIONING PACKS……………………..OFF
Has been removed from the future before start checklist draft. It will be part of the start procedure so you can do this “by the book” in the future LEM.