“Don’t care” is easier said than done. Some people are just like that, they care naturally so that’s why they tilt.
One way I found works for me is that if I go into a 2-3 game losing streak, after that I just stand up and walk away from the PC, doing something else. I have a snack, watch a funny video on YouTube on my phone, pet my dog, just something else than staying in front of the PC in Overwatch. Then I come back 5-10 minutes later and I’m always feeling better.