Checking for play in the steering rack is different from checking the tie rods in one key way - you need to make sure the play is actually in the rack itself, not in the joints. Put the car on level ground, start the engine, and have someone slowly turn the steering wheel from lock to lock while you look underneath the car at the steering rack (where it connects to the tie rods). If you see the rack housing move clearly before the car starts responding to the turn - that's a sign of play in the rack itself. Also listen carefully during this - a rack with a gap often makes a metallic sound.
There's an even simpler method. Turn on the ignition (without starting the engine) and slowly turn the wheel in one direction, counting the rotations. Then slowly return the wheel to center. If the wheel keeps turning a few degrees past center without resistance - you've got play in the rack. Tie rods don't usually do this; they give more of a feeling of play when you're wiggling the front wheel side to side.
From my own experience, I remember a similar situation - at first I thought it was the tie rods, but it turned out the rack was so worn it needed replacing. Worth checking when the steering system was last serviced on your Sportage - usually at this age, tie rods get replaced more often than the rack, but if you've put a lot of miles on it, the rack might need attention too.