It has been a long-standing frustration of many that reservations are not a part of the culture in the neighborhood!
For parties of six or fewer, it can be hugely frustrating! I did find some years ago that, while you may not be able to make specific reservations for six or more, you stand a much better chance of getting seated witihin a reasonable time than were you a party of four or fewer!
A few country clubs may honor a reservation for lunch. Having recently researched this, I found that Mallory Hill Country Club will make reservations for "sizable" parties, but only before noon. On the other hand, Palmer Legends will accept reservations for a "large" party after noon.
I agree that groups for dinner are difficult!
SWR