Does it make any sense to approach recruiters (via LinkedIn or email) of a company X after failing the resume screening of same company X ? Could the outcome be different than applying on company's website ? TC: None
The don't people have no idea what they are doing. Find recruiters on linked in and message them. It's better to get a referral - repher.me is great and free. If your resume gets filtered without a phone screen or anything it means you didn't work hard enough, or are completely unsuited for the role (i.e. 2 yoe undergrad applying for staff or L7 position).
You're better off contacting someone on the actual team/hiring manager to get referred if you can figure it out. Contacting a recruiter might just get routed to the same same recruiter who rejected your resume.