You probably know that using liquid hydrogen to cool engine bells is common in rocketry as well. I found this video fascinating. On one side, the engine is so cold icicles are forming. Centimeters away, the other side is so hot that it would melt pretty much any metal except the most exotic of alloys. The engine is so efficient that all that energy is pointed straight down. None escapes to melt the icicles.
Yes, it keeps the bell of the nozzle from melting off the back of the rocket. And that is a really cool trick, extremely tough to pull off in the original F1 engines of the Saturn V (lots of welds) but easier to do now that you can 3D print the engine.
However, the SABRE has a different challenge. Rather than cool metal (Inconel) which is a very good conductor of heat, the SABRE heat exchanger is cooling the air which has gained temperature for becoming compressed by the moving aircraft. Air, for all of its wonderful properties, is not a particularly good heat conductor[1]. Further, since heat transfer by conduction is a function of time in which the two dissimilar temperature materials are in contact, and hyper sonic craft are going very fast, it has to achieve this heat transfer in a literal "blink of an eye." It has been compared to shooting an acetylene torch through it and having a cool breeze come out the other side. Really amazing stuff.
[1] This is fortunate as it makes down coats warm rather than cold.
Sure doesn't seem that way to me. Blue rocket flame is coming out, and the operators talk about "ignition" and percentage thrust and shut down (when all the icicles fly off).
https://youtu.be/4eM1mNNdguA?t=89
Scott Manley has a good explanation but I can't find the video right now where he mentions it.