other than doing as she suggested, or completely removing the soil in that entire area, I don't have any other suggestions. However, generally after a year, depending upon the species of palm, most of the seeds should have germinated by then so it won't be an indefinite problem.

I agree, it's not difficult to manually remove the seedlings. Probably someone dumped some palm inflorescence loaded with ripe fruit/seeds. That went through a mulcher and the seeds being small and hard would have mostly got through undamaged. Definitely only a short term problem.

Sorry, it seems like you may be over-reacting. Depending on where you live, and what species of palms grow around you, you may have palm seedlings coming up indefinitely - mulch or no mulch. You won't be able to control nature, and so if there's a tree nearby that produces seeds and those seeds find a way to your yard, they'll sprout and you'll just have to deal with it. Whose to say the seeds weren't already in the soil and not in the mulch?

Where I live, washingtonia and phoenix canariensis seeds are dropped by birds and they sprout by the hundreds. Syagrus rommanzofiana reseed in my flower beds also, although they're not as prolific as the former two. I've read that archontophoenix and roystonea are prolific seeders in landscape situations also. If any of these palms is in your area, the issue is probably not the mulch.