Flamestrike
Legend
The whole "deciding which PCs can and cannot roll" thing is absolutely house rules and not a typical reading of that passage.
No, it's not a house rule. The rule expressly states that the DM determines who can roll and when and what the DC is if a check is allowed. Thats the exact interpretation of that passage. It does not say 'once you set a DC every player is then allowed to check against that DC.' The passage does not even infer that interpretation.
You (the player) dont just get to pick up the dice and roll.
Your interpretation leads to results that you yourself state to be absurd. Mine doesnt.
There is a rule of Statutory interpretation whereby if two different interpretations are open to you, and one leads to an absurd result, the other (non absurd) interpretation is to be preferred.
Try it one day. It encourages skill diversity among the PCs, and highlights different skills and ability scores moving the spotlight among the players.