Why do they use a drag chute when they are able to land safely at Edwards and Kennedy without it?
They didn't use a drag chute until 1992, years after they had already lost 7 astronauts to the shuttle. SOAS was first set up for use in 1988, during the first flight after the challenger disaster, but only at the TAL sites. So you didn't answer the question of why they use SOAS at TAL sites but not at KSC or Edwards, and perhaps even more telling, why for those 4 years between 1988 and 1992 didn't they use SOAS as KSC or Edwards but they used it at TAL sites? If you read the challenger accident investigation report you'll find ample evidence that the investigators had concerns about the shuttle's ability to brake in time, especially at abort sites. Yes the drag chute helped their safety margin eventually, but its main purpose was to suppliment braking capability at Kennedy, which the investigation decided needed improving. Despite that, they did indeed land at Kennedy without either a drag chute or a SOAS net after challenger. They were not nearly as flexible about the abort landing sites, and they made sure that SOAS would be implemented at TAL sites before they went to fly again.
From the recommendations section:
"Landing Safety. NASA must take actions to improve landing safety.
The tire, brake and nosewheel steering systems must be improved. These systems do not have sufficient safety margin,
particularly at abort landing sites."