So, lwtsay the 6 Strength Wizard tries to qualify for the Olympics. He doesn't get a d20 roll, because he cannot succeed due to lack of proficiency. DC doesn't come into play. Of a character cannot do something, they can't roll a d20 Test. If they can roll the test, then they have a 5% minimum chabce of success.
So, basically, the only way for it to work is by accepting that a given task has a different TN for every character? I am all for it, but I didn't feel it was the expected way to do it in 5e (even with the disappearance of DCs sample for skills, in many adventure the DC is linked to the task without consideration for who is doing it, based on the assumption that a roll would be done with adventage if a character is especially suited to that (RotFM for example, does that a lot).
I am all for "Looking for notes in a library? Everyone, the TN is 30. The note could be handwritten anywhere on the books. Bob, your background as a Sage has made you familiar with exotic classification systems, for you the TN is only 20. Jack, your character doesn't know how to read draconic, err, you can't roll (the TN would be over 30 for you, which is equal to being impossible)." But deducing this is the way it will work through the reading of two sentence is a lot to assume, including the one that is in question here, that a roll is warranted when the TN is over 5 and under 30.