There are a couple of more downsides for second-curtain sync not yet mentioned in other responses.

With TTL, flash power needed for correct exposure is determined by measuring reflection from a preflash before the shutter opens. Using rear-curtain flash, the distance between subject and the flash may have significantly changed during exposure and the pre-calculated value is more prone to be wrong.

Similarly, focus (especially auto-focus) is usually based on where things were at the start of exposure, not where they are expected to be by the end - you have more chances of missing focus for a moving subject using rear-curtain flash unless you have planned for it.

There are a couple of more downsides for second-curtain sync not yet mentioned in other responses.

With TTL, flash power needed for correct exposure is determined by measuring reflection from a preflash before the shutter opens. Using rear-curtain flash, the distance between subject and the flash may have significantly changed during exposure and the pre-calculated value is more prone to be wrong.

Similarly, focus (especially auto-focus) is usually based on where things were at the start of exposure, not where they are expected to be by the end - you have more chances of missing focus for a moving subject using rear-curtain flash unless you have planned for it.

EDIT: Also, with most cameras viewfinder is black during exposure. Handheld, you will have harder time trying to keep desired composition and avoiding tilting the camera until flash fires.