As the podcast says, mounding mulch around a tree trunk with the high point of the mulch mound at the tree trunk is exactly what one should not be doing. The doughnut shape of mulch, with very little to no mulch at the tree trunk is what we should strive for. As Sooby says, it's important to know how high the root flare is, since it may have purposely been planted above the existing soil surface (as Shadegardener says). This would also determine the overall shape of the mulch mound.
But you are right: don't remove all that mulch and kill the existing roots outright. That would be very stressful for the tree. I would probably remove some, though, to encourage deeper roots and roots elsewhere. It sounds like there might be other complicating factors, like either very heavy or very sandy native soil in your yard, or undesirable watering practices in the the past. But without any more information, that's only speculation.
