mgmsteven

Most of my tatl award flights have been on LH. ATL, IAH, and DEN usually have plenty. You will do better searching for two one ways than a round trip. Go to the award calendar and search for all the dates available on the out bound. Then go back and search for all the dates available on the return. Pick a combination you can live with. Sometimes you may have to lengthen or shorten your trip by a few days to get something you like. I have found that by searching for a round trip for a definite # of days often results in no availability. Simply adding or subtracting a day or two or three will result in availability.
I don't think the Chase Saphire offers extra availability. Being Elite or having a United credit card really adds significant availability. It would definitely be beneficial to have the card plus you get a sign up bonus miles.