Yes I've done this on many of my own vehicles even headlights and usually just wire to an ignition source like you mentioned. If they're LED's you don't really need a relay because they don't draw much but if halogens you definitely need a relay as they can draw 10 amps a piece so a 30 amp relay should be good.I don't think the vehicle has a fuse specific to the fogs, part of something much larger. Same problem here. Have you tested for 12v at the wire right before the bulbs? Mine only shows 5v, so I figure its the stalk switch, didn't want to spend money on that so I just ran power and added a dash switch in one of the blanks for the fogs. Can add a relay to power up only when vehicle is on to save the mistake of leaving them on, but that is kinda obvious, at night anyway. Luck.
I had the same problem... end result replaced this switch for my Optima: Kia 93410-2M110, Combination Switch. Sells for $41 on Amazon
I did everything you have done with the same results.
To Troubleshoot you can try this, with the headlights on jiggle/play with the fog light switch with the headlights on and look for the fog light indicator on the dash to illuminate.
Also, know it took me a lot of attempts of playing with the switch while driving at night before i got the dash indicator to light up.
It took way more than 5 minn of playing before i got results, so keep trying.
I think there are other threads for this same issue - Bad Switch
I have a 2011 Optima and the fog lights aren't working. However, if I move the control stick forward about a 1/4" and then down about an 1/8" of an inch the lights will come on but the stick has to be held there to keep the lights on. Not exactly ideal for driving down the road since if it's...www.optimaforums.com
I have a 12sx Turbo with Technology & Touring package this was the switch for my Optima