TL;DR - The possibility of Small Claims + attention to detail is all it takes, you rarely have to actually file and appear.
I've done this with Sprint in the USA, and later with LG and HTC in the UK (they call it 'Petty Debts' but it's basically like Small Claims in the USA).
Bottom line is that these stickers are effective - much like the parking lot signs that say how the lot owners 'accept no liability' for damage to your car. Whether they accept liability is irrelevant, you can still sue and a court may assign liability whether they like it or accept it or not. This point is overlooked by many who don't want the hassle and that's why such signs (and stickers) are effective.
They are just part of a layered defense, and anyone with the will to pursue a claim can still prevail. It's not as if you show up in court and they point to the damaged sticker on your laptop and the judge says 'case dismissed' or whatever. They have to actually defend their point that their warranty does not apply because of some silly sticker (it almost certainly does).
LG, Sprint, and HTC all did the right thing ultimately when I sent a properly formatted business letter demanding that they uphold their end of the warranty terms, and giving them a deadline of ten days to comply or they were going to be sued in Small Claims. I kept it short and sweet, and attached a comprehensive time-line with exact times, dates, and names of everyone involved's correspondence and what they said. It was a bit of a hassle but still cheaper than repairing or replacing an expensive handset off-contract.
They'll usually send an NDA that they want signed before they pay out, and you can ignore that and let them know the ten-day clock is still ticking. They may also ask you to log in to their support website which subjects you to additional T&C's more restrictive than your legal rights - avoid that, it's a minefield! Stick to registered postal mail sent to the companies registered business address.
I think they assess what it will cost to repair versus what it will cost them to show up in some random county court, and cut their losses when it becomes clear a customer has no intention of giving up. Having complete documentation plus a time-line with everyone's names that you spoke to is the single most effective tool to win when you are up against a big company, so always plan offensively from the start of any trouble with a device and keep a simple text file that you maintain for reference.
You shouldn't do any of the above if you really did break the device by doing something stupid, just take the loss and buy a cheap or refurb replacement. Don't try to lie and plan on them not showing up and you winning a default judgement. But if the company is doing you wrong, by all means go after those bastards.