I don't think you can be an asylum seeker and cross illegally with the zero tolerance policy. I'm assuming those cases are very rare and those people must have very serious, very credible claims.
With the zero tolerance policy, once you cross between the ports of entry illegally you have become an illegal entry, will be prosecuted and forfeited any asylum claim you may have had. There's a right way and a wrong way.
Crossing the border illegally and then pulling out the asylum card once Border Patrol catches you isn't the right way. Say for instance Border Patrol fails to catch them, do you honestly believe they willingly and voluntarily turn themselves in and go through the process? Or, do they disappear into the country?