not going to argue about core crypto primitives - i have no idea what it takes to be able to contribute meaningfully there.
but I have no formal education. and when I started there were no gatekeepers. I mean there were, the MIT grads got to work on much cooler stuff than everyone else.
since then I've collaborated on chip designs and written a metric shitton of low level software - including microcode and alot of parallel synchronization primitives. and when I work with new grads (incl phds), I don't expect them to know any of that really - and we teach them. some of them get it and some of them don't.
not only do I find your emphasis on the 'proper training' misguided - I think its counter productive. unless you're working with a very rare professor, they aren't going to really have a very adequate notion of what new designs are like at all. most of them have had only the most casual experience with industry. your utility to me as a future systems programmer has alot more to do with your general level of talent and your interest in the topic than whether or not someone made you do dining philosophers.
- For low level hardware grovelling, you need a sufficient understanding of the workings of modern chip internals.
- For crypto, another "don't invent your own" thing, probably a specialised degree in cryptography and a lot of practical experience.
The bar is simply higher than "I have access to a programming language and think my idea would work".