I'm not sure where you got this info, but it's not what happened with me. I have a fulltime permanent position which I got without the LMIA. I told my HR I wanted to apply for PR and needed the LMIA. They filed it on 1/10 and I got the positive LMIA returned in the mail marked 1/30. It will give me a 1 year valid work permit.
I recently filled out the application on EE and stated that my current permit has no LMIA, but my job offer has a LMIA. I was awarded the 600 points and am listed as qualifying for the FSW and CEC.
Just to be on the safe side, my employer has gone a head and filed online for a new work permit based on the positive LMIA. Once this comes through I will update my profile saying that my current work permit has a positive LMIA. The lawyer said this is fine.
That being said, I realize that I am extremely lucky to have an employer who is willing to file all this paperwork for me. I'll give updates on what happens if anyone is interested.