Are supports supposed to pick first?

In every game I queue into, someone is bound to request to switch pick times with me if I’m a later pick, right as we get into champ select.

I’ve had games where people would get genuinely angry if I don’t swap with them even when they already see who they’re going against and I wouldn’t if I swapped.

I always try to counter pick the enemy support if I can so I would really like to know if I should be letting other lanes pick first instead?