DHS employees by and large cannot be bribed. It is a serious offense, and both the receiver and payer will be brutally punished. So immigration officers have literally no incentive to help you. Nothing good happens to them if they process your case or help you. Something good may happen to them if they brutalize you because prosecutions are good for their reviews for promotion.
Whenever things end up like this it is good to take a step back and realize people by and large are people. The guy in Honduras letting someone in for slipping a $20 isn't much different than the CBP guy in America who ships a guy off for CECOT for having a soccer tattoo.
You can't fix this system until there is something in it for the officer enforcing it. They need some mechanism for legal bribery, like a reward for letting in and keeping good people or to just straight up legalize people paying off immigration so that normal people get all the benefit drug traffickers already do.