How do I demonstrate that I am seasoned in a way such that if a Big Tech company decided to give me a chance, I can put them at ease and show that I'll be successful? How does that look like across various levels?
For non-senior folks like yourself, the best thing you can do is show very strong communication and coding skills. For Big Tech, this will generally be through the data structures and algorithms (DSA) rounds.
Even though DSA is far from real production code, you can still show a lot of strong software engineer fundamentals in these rounds:
When they ask you "How would you test your code?", be thorough going through edge cases. I have seen so many candidates struggle in this portion and only come up with 1-2 cases, because they very incorrectly think that they're done after they just get working code on the board. It's important to remember that at a Big Tech company, missing just a 1% edge case could impact over 10 million users. This exercise should always be taken extremely seriously (and it usually isn't at weaker companies).
Try to come up with alternate approaches with different trade-offs; there usually is a solution that will sacrifice run-time for space usage or vice-versa. This is a huge part about succeeding at Big Tech or any other great tech company: The problems you face there will often times have multiple good solutions that need to be thoroughly examined and discussed. In general, it's absolutely vital that you really talk through and explain your approach.
During my time at Meta, I saw many engineers not make it, mainly due to the above points about code quality. The bar at Big Tech is frankly going to be higher at most other companies - That's why these companies were able to get where they are today. These engineers would come in with very bad coding habits and clearly never seriously thought about code quality and scalability before. This would lead to lots of pushback during code review, which leads to poor velocity, which leads to getting kicked out of the company.
Things do shift majorly at senior (L5+) though. Leadership skills and deeper behavior becomes much more fundamental. System design as well. But I wouldn't worry too much about that - Just focus on really knocking the DSA rounds (or practical coding rounds) out of the park.