The fifth edition of D&D is very particular about not spamming spells of level 6-9.
This is most obvious when it comes to warlocks, who have enormous freedom in what spells of level 1-5 they can cast and when, but who can only cast spells of levels 6-9 once a day. Their special features that give them more spells after a rest only work up to level 5.
Similarly, sorcerers can only use sorcery points for spells of level 1 through 5. Minor spell-casting classes only cast spells up to level 5. Monk's ki abilities only mimic spells of up to level 5. Druids' and clerics' circle/domain powers only grant additional spells up to level 5. A wizard's arcane recovery only works for spells of up to level 5.
Clearly the developers are going to extreme lengths to distinguish between spells of level 1-5 and those of level 6-9. My question is:
What shift in power occurs between levels 5 and 6 that makes it imperative to restrict access to the higher levels?
I've found some partial answers. For instance, no classes have spells that allow for transportation of a large group of people at once until level 6 (when players get Arcane Gate and Wind Walk). Also, level 6 seems to be when all classes get spells forming damaging barriers (Wall of Ice, Wall of Thorns, and Blade Barrier). But this only explains a small subset of spells.
A perfect answer would include why the jump from level 5 to 6 is more drastic than the jump from 3 to 4 or 4 to 5, for instance.