I can't speak for Slack, but it's not unreasonable to believe that a single machine's available output bandwidth (~10-40Gbps) can be saturated during a deploy of ~GB to hundreds of machines. Pushing the package to S3 and fetching it back down lets the bandwidth get spread over more machines and over different network paths (e.g. in other data centers)
We do it similarly except we push an image to a docker registry (backed by multi-region S3), then you can use e.g. ansible to pull it to 5, 10, 25, 100% of your machines. It "feels" like push though, except that you're staging the artifact somewhere. But when booting a new host it'll fetch it from the same place.
