Exactly. This has been on the way for a long time, and popular resentment had begun to brew as many of the "noveau riche" had taken to flaunting their extra children as a status symbol -- the fine levied for an extra child was exorbitant if you were a rural peasant, but not that big a deal (a few thousand dollars) if you were one of the urban elite.
I wouldn't exactly agree with the "two thirds already exempt" bit, but there were exemptions for the various non-Han ethnic groups in China (which are only about 8% of the total population). The other exemption allowed to many was a second child if the first was a female (enacted in part to prevent mass infanticde of girls and that was only since 2007). There was also a provincial-level decision made in 2007 (2011 in Henan) that if both parents were only children, the couple could have a second child.
In 2014, Zhejiang allowed for a 2nd child if even ONE parent was an only child, and since then all the provinces except Tibet and Xinjiang have followed suit (where large portions of the population are non-Han and thus not subject to the policy anyways).
What they've found is that a lot of people aren't option for a 2nd child even if allowed, because damn -- kids are mad expensive, yo.
There's also a looming demographic bulge created by the one-child policy. It's been in place for 35 years, and if you think of it as an "anti-baby boom", the repercussions are apparent. You have a large segment of the population born prior to 1980 which are hitting middle and old age, and a considerably smaller proportion of young people in China to work and pay taxes and to support all those old Chinese.