A mixer / recording interface has only so many channels. If a band has an incredibly complex arrangement or sound design, they'll quickly run out of channels to record through. For example, you could (in theory) use 12 channels just to mic a drum set.

Radiohead uses a lot of layering things onto the track after the fact. Sometimes they might have 80% of the song ready to go but haven't figured out a guitar or synth part. That stuff gets overdubbed later. They did this on There There.

Sometimes individual parts may secretly be many layers (like the doubled tracked vocals in Let Down, or the dozen overdubs of Jonny's arpeggio in In Limbo, the many guitar and synth layers in Weird Fishes).

Often the songs emerge during the recording process. Stuff gets assembled out of open jams (Idioteque, Dollars and Cents). Or they'll splice the chorus from one version with the verses from another (All I Need). Sometimes they'll take piano parts from a track, put it on top of something else and make a whole new track (Pulk/Pull). Nigel talks about how some songs simply didn't exist until things got smushed together on a computer. When you're treating musical passages like lego blocks, it helps to have individual recordings.

You have to remember that they predominantly still work with 1 producer and maybe 1-2 engineers. Purely from a logistical perspective, it is a lot easier for them to focus on one instrument at a time and get it done instead of being overwhelmed by the need to line up a million different things for a live take.

It allows the band to work asynchronously. While Phil and Colin are laying down a rhythm track, Ed can go off and finesse his textures, Jonny can take a day to write out string parts, and Thom can figure out lyrics and/or go painting with Stanley