What's the purpose of SMA antenna connector grounding? Can I just use a copper antenna with no grounding?
Every antenna feed point requires two nets. If it's an unbalanced feed, the second net is the ground return.

By connecting the ground to the SMA connector, the board designer gives you the flexibility to use different antenna implementations. You may have a cable and then a remote antenna, or you may have an antenna right at the connector.

If you have a monopole antenna, you need to have a reference ground, which could be implemented using the board's ground plane. If you're simply going to connect a monopole antenna right at the SMA connector (no cable), then you may be right and the ground connection on the SMA side may not be strictly necessary, since the board ground will be part of the antenna. However, I see no penalty in connecting the SMA ground even in this case. You may actually be saving yourself from some trouble: the antenna you're going to use may have been designed taking the length on the SMA ground into consideration, or there may be matching components connected to the ground inside the antenna.