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do you have an itemised bill for the 1st service?
as there should be a line in there that state' check for full service and safe operation of braking system'
If the slider had rotted ( and I doubt that this has happened) it should have been picked up on that part of the service
I'd ask to see the 'rotted' and broken parts as I have never seen a brake slider thats corroded - usually they are made of stainless steel and so cannot rust.
what I suspect is when they replaced the pads they failed to tighten a slider bolt or cross threaded it so the slider then falls out or comes loose which then locks up the caliper and grinds the pads and discs down.
so again I say, ask to see the broken / rotted caliper parts
I've attached an image of a typical brake caliper - yours will look something like this
the 'sliders' are parts 11 and 9 on the image so hopefully you will get a better idea of how it all works