So here is my analysis of the process, FWIW. I "Kick" the pedal to get the toe to go into the strap (back of the pedal) and as I slide my shoe in, the cleat pushes down on the rear part of the cage (due to the ramp of the cleat) forcing the front of the pedal up with a result of the toe of the shoe hitting the rear edge of the front cage. Without the "inviter" it gets hung up on the vertical rear surface of the front cage. With it, the toe slides on in while the cleat moves over the edge with the final movement being the rear cage sliding into the cleat slot. How well this works is a function of how tight the strap is and/or how much the clip is open. Both cases represent pressure on the top of the shoe impacting the level of ease of the shoe insertion.

Different shoes can result in different ease of insertion. Mine are old Avia's with Velcro flap on top making them a little thick.