Extremely late Letter of Recommendation. Hurt my chances?

[JidneyKidney04]

I applied to 9 different PhD programs in developmental psychology. I have 2 LOR in with one missing on the apps that were due early December. 

I asked the professor a month in advance and he seemed very happy to do it. But I reminded him a week, then a day before the deadline and still no letters were submitted. I went to his office on the 1st when most apps were due, and he seemed annoyed and said he was "all tied up" (which I can understand since it was nearing the end of the semester). He also said LOR from professors technically don't have to be in when our apps are due anyway. But, it is now the 22nd and his letters are still missing.

I got interview invitations from two of the schools I applied to despite his missing letter, but the other 7 are top schools (most being UC schools). I see other applicants are already getting invitations from those programs.

I have good GRE scores, great background in research and also did research at UCSD (1 of my LOR is from a prof. there too). I feel I had a good chance at at least getting an interview, but I'm not sure how much a missing LOR weighs. Without one of the 3 needed, I feel my applications suffer a lot considering I don't have PERFECT scores/GPA and outstanding research experience/publications.
My question is, are applications simply tossed with one LOR missing, or do adm. offices send notifications of a missing LOR before officially tossing the app? Do top schools even look at apps with a LOR missing?

What should I do since my professor is not responding and I have no one else to ask to replace his letter??

[TakeruK]

Have you talked to the school that you are applying to yet? Find out when the absolute latest deadline is. Many schools may have deadlines in December but they won't review the applications until January, so as long as the letter is submitted before then (and that they know to expect it late), it should not make a big difference.