Yes, the ignition switch is under the steering column
What you changed is the ignition lock cylinder, no wiring there, well 1 wire for the "key in lock" dinger, lol
Simplified drawing here of the way it works:
https://www.2carpros.com/forum/automotive_pictures/62217_berretta_1.jpg
The key cylinder will only turn if correct key is in it
When key turns, a gear on the back, the selector, turns and pulls(or pushes) the actuator rack(rod) that runs down the inside of the steering column
Ignition switch has a Post on it that the actuator rod is attached to, as the key is turned the ignition switch is moved into its 4 positions, ACC, OFF, RUN, START
Turning the key to RUN or START also unlocks the steering wheel, also shown in drawing
Yes, pull out the ignition switch and then grab the end of actuator rod and rotate the key to see if something is loose
Close up of the gear setup on back of key cylinder:
https://static-resources.imageservice.cloud/3794807/key-spins-in-cylinder-wont-start-ford-truck-enthusiasts-forums.gif
Either one of those gears can strip and would need to be replaced
Picture of ignition switch here:
https://i.ytimg.com/vi/KK2uO_xaDKA/maxresdefault.jpg
You can see the post that the actuator rod moves to turn on and off systems in the vehicle
In the movies or TV shows when "they" reach under the dash they are grabbing the ignition switch and pulling it out, so they just need to slide that post to start the engine
Problem is locking steering wheel, lol, but its the movies, so I guess those cars are special