Why do all these scattered dates fall on the same weekday each year?

They're deliberately spaced whole numbers of weeks apart, so they always share one weekday — the doomsday.

For example, 4/4 to 6/6 is exactly 63 days = 9 × 7, so they're always the same weekday; 6/6 to 8/8 is 63 days too. The odd-month and February anchors are placed to line up the same way. The set isn't magic — it's a carefully chosen collection of dates that are all a multiple of 7 apart, which is what lets one "doomsday" cover the whole year.