Russia launched about a dozen Proton rockets a year at the time of Columbia disaster, two Soyuz spacecrafts and two-three Progress spacecrafts.
Proton used to launch several Soyuz without Orbital Section, called Zond, in 1960s, with booster DM to fly around the Moon. DM adds about 3 km/s of deltaV. Orbital plane change for 7 degrees - between Baikonur's 46 and Columbia's 39 - takes sin(7)*8 km/s = 1 km/s. So technically it could be considered to launch (several) Progress ship(s) with necessary cargos to Columbia while three unmanned Soyuzes would be prepared and sent to the rescue. Soyuz approaches Columbia, gets caught by manipulator, astronauts use spacesuits from Soyuz - which have to be extracted from Soyuz first.
ISS, meanwhile, had to be kept without ships, which means landing the crew, hopefully temporarily.
All that could be considered. But I don't think anybody high in chain of command considered all of that necessary.
I don't think Columbia had a docking collar, did they even have spacesuits on board? If not, they wouldn't have been able to get to the Soyuz to extract anything. You'd have to send up at least one person one the Soyuz to pass the spacesuits into the shuttle airlock.
Also, I'm not sure Columbia had a manipulator arm installed at the time. It's heavy and they don't fly with it when they're not going to use it.
Yes, there are problems. Soyuz could fly pretty close to Columbia, but to get to the bay could be tricky. It was not going to be a regular docking - but something for a saving mission.
If Soyuz could be taken to the Columbia bay, an astronaut in the Columbia spacesuit could approach Soyuz and open the hatch. Spacesuits could be extracted from Soyuz without entering the spacecraft. Talk about ingenious ideas with rope, attached to the front hatch, etc. The Soyuz spacesuit then is delivered to inside Columbia, where somebody could wear it to get to Soyuz.
Proton used to launch several Soyuz without Orbital Section, called Zond, in 1960s, with booster DM to fly around the Moon. DM adds about 3 km/s of deltaV. Orbital plane change for 7 degrees - between Baikonur's 46 and Columbia's 39 - takes sin(7)*8 km/s = 1 km/s. So technically it could be considered to launch (several) Progress ship(s) with necessary cargos to Columbia while three unmanned Soyuzes would be prepared and sent to the rescue. Soyuz approaches Columbia, gets caught by manipulator, astronauts use spacesuits from Soyuz - which have to be extracted from Soyuz first.
ISS, meanwhile, had to be kept without ships, which means landing the crew, hopefully temporarily.
All that could be considered. But I don't think anybody high in chain of command considered all of that necessary.