I am on my STEM OPT, and it is valid till July 2024.
My H1B was picked(March 2022) and approved(August 2022) last year.
However, my H1B was approved under the Consulate processing and received ( I-797 B). So, I asked my employer about it and got to know that I can change to H1 status whenever I intend to or after my stem opt expires.
I am planning to go to India in Oct for stamping. So I asked my employer to change my status to H1B, and they mentioned COS is not needed, I can directly go on STEM OPT and get my H1B visa stamped. It will automatically change the status to H1B.
Your employer is correct.