If their recruiter is "super impressed with my resume" and thinks I'd make great things happen at ${COMPANY}, then why can't they enter my experience into their proprietary system?

I mean, they were so confident in their initial email to me. Were they not actually as amazed as they stated?

Most job applications are not five minute processes, even for day laborers. Source: me, doing payroll for thousands of staffing companies.

If the 5 minute form got me a job, sure. But it's a 5 minute form to chuck a bunch of my personal information into a black hole where, most likely, I'll never hear anything, or will get a form letter back in two weeks telling me they're not interested in me for the role I already have at a different company.

So really, as OP put, it's 20-50 5-minute forms that are all exactly the same, so that you might get a single interview. Or you just know someone who works there and skip the whole thing because woo nepotism.