Not quite. You have to deposit a copy of the manuscript in PubMedCentral within a year of publication. Some journals let you deposit the final pdf (or even do it for you). Others restrict you to an unformatted "final draft" (i.e., the last thing you submitted to the journal).

However, this is still tagged with the source journal, so it doesn't really get you around the career-related issues.

Public Access guidelines: https://publicaccess.nih.gov/policy.htm