No – that wasn’t the reason for the flat tire. The flat tire had a cause but not all causes are reasons. You didn’t arrange for the flat tire, either – so you didn’t provide a forward-looking reason. You simply reacted to what happened – you made lemonade from the lemons.
I’m with you on cringing when I hear «everything happens for a reason.» But I don’t think you’ve solved anything here. There is no fix for the phrase. It’s just that some people feel uncomfortable with the fact that much of life is out of their control.