In Colorado, the key thing they look for is "progressive engineering experience", meaning you need to do more than just plan review, CAD work, and quantities. The PE oversight isnt as critical or required, but it is preferred. The important thing to highlight on your application is that you have grown as an engineer. I know some engineers who spent their first 5 years out of school only estimating, and the board required almost 3 more years of experience. Even I needed an extra 4 months of experience because I had a relatively flat start to my career.
IMO, its not about who your employer is, its about what you do while you work there and to a certain extent how you present what you did.