They specifically mentioned (and showed Nhan setting up) transport signal enhancers. They also told Reno that using them was a backup plan in case the Hiawatha's own transporter didn't work.
Burnham: (On the way to the pods) We have less than two hours to fly
down to the asteroid's surface and search for survivors then set up pattern enhancers and beam back.
On the asteroid Burnham discovers the Hiawatha's transporter is operable.
Burnham: Set up the enhancers around the ship, widest perimeter possible, then meet us at Sickbay.
Nhan: Got it.
Bunham: We're in business.
Reno: Why are we moving them if you're creating an enhancement field big enough to beam from anywhere on the ship?
Burnham: The enhancers are just a backup in case the Hiawatha's main transporter doesn't stay online. Ship to ship beaming on the pads is far more reliable and will be safer for your patients.
So the use of Haiwatha's transporter pads was specifically to reduce risk, especially for the critically wounded survivors, but otherwise once the enhancers were up, Discovery could beam people down and up with an acceptable risk.