ps, I've never actually had a cut tire, ever.
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I had several sidewall failures when I was in high school. I think the bike shop had a bad batch of tires. After two or three new tires blowing out in a week or two, the bike shop telling me that that was their best tire in that size so it obviously was my fault, I bought their cheapest tire and stopped having blowouts.
I had a glass cut across the tread on my rando bike. I was not sure if enough of the cord was cut that the cut would grow in size, so I booted it with tyvek. A year or two later replaced the tire so I do not know if the tyvek was needed or not, but I certainly do not regret using it.