We don’t often light fires nowadays and the everyday words describing the process are being forgotten.
For example:
K. L. set alight to 72-year-old J.C. in her
garden [...], on 13 March 2018 before killing himself.
[bbc.co.uk/news]
A man has been injured after the car he was in went on fire on an
unclassified road between Dingwall and Evanton. [bbc.co.uk/news]
You look at me and, babe, I wanna catch on fire [Lady Gaga: Always
Remember Us This Way]
Three people were onboard when it came down, hitting power lines and
setting alight on the A14 near Abergavenny. [BBC TV Ten O’Clock News]
They tore down a giant banner celebrating communist China's 70th
anniversary and rapidly put it to flame. [BBC TV Ten O’Clock News]
...to light on fire [ChronicleLive.co.uk]
You can light a fire.
You can set light to something, set fire to something, set something alight and set something on fire.
You can’t set alight to something. Nothing can set alight.
Things catch fire. (In some dialects you might wait, say, for the wood to catch.)
Once they have caught fire, things are on fire. But they don’t go on fire or catch on fire or light on fire.
While "put to the sword" appears in print occasionally, "put to flame" and "put to the flame" make very rare appearances: mostly in the writings of Lord of the Rings fans.