Is this a structural issue with how recruiters are compensated?

At many companies they have quotas or are paid bonuses based on how many candidates they source. I can imagine a policy whereby candidates that approach the company directly aren't counted as "sourced" by any particular recruiter.

So far out of all the forms I've filled out, Square is the only one who actually responded via the forms. It led to an offer which I turned down for other reasons but it did work that one time.

Except for that offer, I get all of my jobs through boutique recruiting firms and some larger recruiting firms or connections.

For what it's worth, I've gotten a response to an application to Stripe specifically from applying through their website. No experience at FAANG or other high-profile companies. And I had previously applied for a different role (with no response) and the recruiter was aware of that.

My impression is that the main issue is stale listings - you probably applied for an opening that they were effectively done reviewing applications for but hadn't taken down from their website. One recent user-hostile trend I've noticed in job application platforms (including Stripe's) is that they don't tell you when a listing was posted or updated. So you can't distinguish between an opening posted yesterday (that they're probably actively reviewing applications for) and one posted six months ago (that's almost certainly a black hole).

My experience getting a job in the last few months is similar; I got a response from almost everything I applied to. In most cases I didn't even bother writing a cover letter (I hate writing those so much, and it takes forever).