This is an interesting question to explore in concept. Or at least, I find this concept somewhat interesting to explore.
In general, where would one get data on the failure rates of mechanical or electronic equipment? Frequently, there is no government-mandated reporting for failures of, for example, hard drives. In public health, my main field, we would be able to get mortality information from, for examples in the US, the Social Security Administration and Medicare's administrative data. I'm not sure what the situation is for automobiles, but there is a Vehicle Identification Number system that is used at least all throughout the United States. Information like mileage, major accidents, major repairs, sales, and scrapping is reported to commercial bodies like Carfax.
Bicycles lack such a universal identifier to begin with. In the case of hard drives, you will often have a situation where some company owns a big bunch of them - for example, ones that run cloud storage or computing centers. Those companies can conduct reliability studies on their hard drives, as in this example. But again, bicycles are not in this situation.
You might retort that surely the manufacturers have information on failure rates. They surely do, but their information is most properly thought of as warranty rates. If I destroyed a second-hand frame, it is not likely that I would report it to the manufacturer, since warranties almost always extend to the original owner only. Alternatively, if the original owner destroys a thirty year old frame, it's possible they might not report that to the manufacturer, either. They might have lost the paperwork, or forgot that item had an applicable warranty, or just not want to be bothered.
You're correct that bike store reports might be of some use. However, consider that the probability of failure for good mountain bikes is low. Any one store might only get a handful of failures in a year. To get reliable data, you would want to aggregate responses from multiple shops, preferably some sort of random sample from a number of geographic areas. You would also need to specify the reporting period, e.g. how many failures in the last 12 months? A full year would be prone to recall bias, a fancy term for the fact that people can't necessarily recall all the way back to one year very well.
You also lack a denominator, i.e. how many total customers each store has, or how many total bikes are in play. For example, aluminum bikes probably outnumber carbon bikes in total. If you get more failures of aluminum bikes reported, is that just because there are more aluminum bikes in circulation? This a bigger problem than you think.
TL;DR: Manufacturer data are probably the most reliable data, despite the flaws I discussed. It's possible you could combine historical warranty data from several manufacturers and analyze them. The question is how you would convince them to play ball, which I'm not sure I can answer. You are better off answering the question from first principles, like David did.