Generally, if it's a blowout and you didn't damage the sidewall of the tire by slamming into a curb or something, the rental company will usually chalk that up to a tire defect and not ding you for it.

All of that said, it's hard to know how Sixt will handle it. It's certainly not out of the realm of possibility that they would take longer than two weeks to process the paperwork to charge you.

No idea if Chase would honor the claim if you closed the card. If you have the Sapphire Preferred card, it may by worth paying the $95 annual fee just as insurance (no pun intended). If you're closing the Sapphire Reserve to avoid the annual fee, that may be a bit of a tougher call.