Imagine if you knew how to develop databases very well, it was your company's core competency and you were (rightfully) opinionated about it. Now imagine you're interviewing someone, and they show you a database they developed, ostensibly to demonstrate familiarity or initiative with the subject matter. Except it's not a database at all, and it doesn't at all match the specification for what your firm (or similar firms) would consider a database. It's just a peculiar method of storing data in a custom file structure.
If you attempt to develop a trading algorithm independently and without prior training, you are approximately certain to design something from first principles that is utterly unlike what professionals do on a day to day basis. Take a look at /r/algotrading (or similar forums) sometime and read about the preconceived notions people have about that a trading algorithm is and what it takes to develop one. These are programmers who read about the subject matter, not simpletons. It's just that there is far more development and strategy work for the modal trading algorithm to be close to something resembling professional work, and the unknown unknowns are very difficult to eliminate without explicit guidance.
That's not to say it's literally undoable, I am aware of people who successfully do it. It's just risky, because you have far higher downside than upside in trying to go that route and present your work to a firm. The problem is not just that you might not impress the interviewer, but that they may (for better or worse) believe you have a lot that needs to be "unlearned."