It's not 5,166 breakthrough cases in a week, the twitter poster read the poorly worded article wrong. That's the number going all the way back to the beginning.

It was reported that as of last week, there were 5,166 breakthrough cases to date.

This week, it was reported that there were 6,373 breakthrough cases to date.

So there were 1,207 breakthrough cases in the past week which is 38% of total cases in that time period per the article.

That means 30% of the population (unvaccinated) was the source of 62% of total infections.

The vaccines work. Not perfect, but we would be fucked right now without them.