I have answers that do not involve ignition cylinder replacement! I found this question asked in multiple posts but none ended in a solution. Keep in mind, this was a semi-quick proof-of-concept fix with the materials I had on hand. I have no idea if this will work in all cases, but at least it's something.
First, if you don't have time to work all the way through this, the easiest way to get key remote function back is to just disconnect the ignition key insertion switch. The connector is on the left side of the steering column, under the trim pieces about even with the dash pad. In the first picture, it is the one on top; four pin with two wires.
In my case, the problem was caused by stacked tolerances, for lack of a better description; probably a combination of fatigued plastic and slightly worn parts. Essentially what I did is "tighten" the switch with a couple plastic ties. One went just around the switch itself, the second went around the ignition cylinder and switch to remove a little more tolerance and keep the first tie in place.
Getting the first zip tie around only the switch is easier if you can remove the microswitch from the ignition cylinder. Small fingers here are an asset.
The steering wheel needs to come off in order to remove the top column trim, but with the turn signal stalk removed I was able to move the trim far enough left to make sufficient clearance around the top of the cylinder to get a 1/4"-drive #2 Phillips socket and ratchet in there to remove the microswitch. The connector to the vehicle harness will come off of its mount with a small tap towards the front of the car. Once out, loop a small zip tie and snug it down around the top of the white microswitch enclosure. This should force the microswitch inside it down slightly and remove any slop.
Reinstall the microswitch back into the cylinder. Remove the bracket under the column that is held on my four Phillips screws to get additional clearance. Feed a long tie around the cylinder, routing it between the two wires on the switch, and snug it down, making sure that all that is in the loop is the cylinder and switch (no wires, brackets, etc.)
At this point you can test it, and if all goes well, reassemble. In trials, I found that if I removed the key too slowly the key reminder would still stay on, but in real-world scenarios that has not happened since I did the procedure.
Give it a shot, and let me know if you have any questions.