You cannot necessarily trust TIMATIC in these situations, you have to talk to the airlines to confirm. I know that from experience, one time TIMATIC was saying one thing, but all the airlines were categorically saying NO.

In any case, it will probably be best (although not necessarily required) if you can rebook to a direct USA-ATH flight, or at least via EU, to avoid complications.

Another theoretical option, if you have time, is to apply for renewal of the passport from the US via the LA/closest Greek consulate. It takes about a month because it has to go to Greece and back, even if they let you do this on a visa, but another problem is that the visa may expire before the new passport arrives or you may simply not have enough time left for this.

In the end, the US should want you out of the country by the time the visa expires, so they should allow you on a plane back home one way or another...