We had the exact same problems as all you describe. Took it to the shop several times but they couldn't do anything until they were able to replicate it not starting. Very frustrating – but I get it.
Finally, one day we decided to just have it towed to the dealer (road side assistants is great) then they couldn’t get it to start. So they were finally able to replicate it.
The problem was the Break Light Switch. This switch is a relay for the brake lights, gear shifter and starter. They replaced it three days ago and has been working ever since. Here is the write up:
CUSTOMER STATES CAR WILL NOT START WHEN BUTTON IS PRESSED
AT TIMES/LIGHTS ON DASH COME ON/NO CRANK
FOUND B1978. FUTHER DIAGNOSTIC FOUND STOP LAMP SWITCH
CAUSING VEHICAL NOT TO START
UPDATED TECHLINE CASE T1765991
REPLACED STOP LAMP SWITCH AND CLEARED DTC FROM PDM
93810R00 0.3 N94 C15 TT 0.3
PART 93810-3K000 SWITCH ASSY-STOP
I think since they don’t have many cars with the keyless entry this is a fairly new concept to them. My dad works for a Lexus dealership, all models have keyless starting, and said this problem has happen a lot recently. So Lexus and Kia must be getting them from the same supplier.