You can't really "learn to become seniors sooner" if your seniority is measured in years of experience working in a classic enterprise environment.
This is a problem of classic enterprise environment that might be addressed separately.
I personally strongly advise everybody against applying to Big5, or even Big100, until they will come to headhunt you themselves.
In a healthy environment, we look at what the applicant might show (gh/test/interview.)
I agree with your suggested method of evaluation. However, most companies wont even call you back if you don't have some big name on your resume. And, in my experience, a large enterprise has more "wiggle room", for lack of a better term, to hire junior developers. In a startup, there are fewer safety barriers between the junior dev and the production codebase. A large company often has a large amount of non-critical projects that a junior dev can join to learn the non-technical processes that are critical to the stability of the client facing product.
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