No it's not real, I'm currently a tank/caster DPS main, I never feel anxious tanking, not even in Ultimates, not even in Criterion or DRS where aggro and positioning matter.
No it's not normal to feel anxiety over a VIDEO GAME, whether you're casual or hardcore.
If you unironically feel anxious at having to tank, go see a therapist, unironically go see one. It's not normal. It means you are extremely, extremely fragile, so fragile you unironically get anxious at something completely virtual, you'll need to get it fixed.
Even if you suck at the game no it's not normal to feel anxious. Why do you care that you suck? It's VIRTUAL, it does not matter.
