> High rep oly movements = dumb? In what situation?

In all situations. This is never a good idea. Ever.

> With what intended training effect?

If it's to improve technique, do more sets. If it's to improve cardiovascular conditioning, do something else.

> a periodized training program

Oh, you mean the kind of "voodoo science" that Crossfit HQ specificially eschews and that every top level Crossfit Games competitor nevertheless follows?

> If the exercise is stopped when form breaks down, I see nothing wrong here.

I'll say it again: quality control and exercise selection.

> If you've built up to it and have no achilles issues, this is not a concern.

And yet I see middle-aged housewives doing AMRAPs on box jumps.

And it's not just repetitive strain injuries. Misjudge the jump (because, I dunno, you're really tired from high rep box jumping), land on toes, fall down, snap.

A lot of Crossfit is fine. The problems still remain that quality control is explicitly non-existent and that exercise selection is hit-and-miss with a genuine fondness for stupid ideas.

Basically, no good and safe Crossfit gym has any resemblance to Crossfit HQ's vision except to pay a licensing fee to use the trademark.