It's possible that there were no "larger" tables (like for more than 4 people) available for the times you need. When you called, by any chance did the CM ask you if you were willing to try for 2 reservations of 3 people each? You may not want to do that, or may not be able to depending on the make-up of your group, but we have done that several times in the past when a table for 6 or more was not available but our whole family group wanted to dine together. When I was trying to work that out, I called and a CM helped me to get 2 (and one time, 3) separate reservations for the same location (had to be in 3 different names & had 3 different confirmation #'s), and were at times that were close together (I think they were usually within 15-ish min of each other). We all checked in together at the earliest time and explained that "we were together and if it was possible, could they please try to seat us close to each other?" Almost every time they were able to help us, and seated us near each other, and a couple of times they actually did seat us at a larger table together even though our reservations were not specifically for that. The things that I think helped the most are that we checked in at least 20 min ahead of the earliest reservation, asked as nicely as we could, let them know that we were willing to wait a little extra if it would help, and went in knowing that our reservations were not made that way -- and that it might not happen. They do try to help out with seating requests when they can, but sometimes, for many reasons, they just can't make it happen. We did this for a couple of different trips, the most recent being about 3 yrs ago, so I don't know if things have changed too much to do it now, but it's worth asking if it's an idea that would work for you.
Good luck!!! I hope you are able to get some that you will all enjoy!! I will also suggest the usual - keep checking! Even after you get there something might open up.