Nope. The public option on health care did not make the Democratic Party platform.
As Sarah Kliff explains, on health care the platform is mostly backwards-looking: there's a lot more of touting the achievements of the ACA than there are of new proposals for the future. There are some proposals for the future, however...but the public option isn't among them. It's fallen so thoroughly off the radar that Kliff doesn't even mention it.
Has there been any public policy proposal with a stranger path? In 2008, it was a minor point, one among many provisions in the comprehensive plans put forward by all the leading Democratic presidential candidates. It received little attention then.
Fast forward a couple of years, and all of a sudden it is, for many liberals, the absolute core of health care reform, such that when it is defeated eventually in the Senate it's considered by many liberals a betrayal and a sell-out which makes the entire bill a mistake.
I concluded from this that the long-term future of the public option was pretty good. Since it polls well among the general public and had intense supporters within the party, the odds were good that all viable Democratic candidates would support it; as long as that was the case, it would stand a good chance of being adopted the next time the Democrats help the House, the Senate and the presidency, at least assuming that the rest of the ACA was in place. After all, the public option can easily be grafted on to what would be in place then, and it can be passed on it's own through reconciliation. No need for 60 in the Senate.
And then...nothing. It's as if it never existed. Leading Democratic Senate candidates this cycle did not put it in the issue section of their web sites, even in liberal states with contested primaries where one would think that candidates would be looking for anything that appealed to liberals. And now it's not part of the party platform, and as far as I can tell no one really cares (Jon Walker at FDL does have an item today mentioning it; a quick search yields pretty much nothing else, and if there was any campaign to get it into the platform, it certainly didn't make very much noise).
I can think of lots of cases of an out-party caring passionately about an issue and then dropping it when they win. For a party, or at least the activists in a party, to begin caring passionately about an issue only after they are voted into office, and then drop it in time for the next election...that's weird. I don't really have any conclusions about it. I'm not sure I even have a theory.