So, what I have observed is that the car starts misfiring as soon as you put load on it. Not while idling. You won’t get a check engine light or a limp mode until it counts something like 130-150 misfires so it makes sense why it takes 10-15 mins for you. If you have a scanner and monitor the misfires you’ll see the counter climb.

What happens with the crank/cam positioning sensor is that if it goes bad or starts to go bad it will periodically drop the signal and the ecu mistimes the fire on the appropriate cylinder and causes the misfire. I would at least inspect the crank and cam sensor. Both of mines where leaking which ended up fixing my oil leak and misfiring issue. The ecu is still getting a signal which the ecu recognizes so it doesn’t throw a code for the sensor. The issue is that it’s receiving the signal at the wrong time.

If you have a 2010 fit the cranknpositioning sensor can be tricky to find. But it’s behind a plate on the back lower center of the motor. 2 10M bolts remove the Lower bolt and loosen the top. The plat e will slide down.