"Without fail, 1, 2, 3, 4 or 5 riders make a breakaway and typically open up a lead of 3.5 minutes. Never seems to vary much from that.
The peleton maintains exactly the same average speed throughout, and again without fail, gobbles up the breakaway with 500-1000m left (never less, never more) and the 5 guys wind up coming in 20 seconds behind the rest.
So the question is, why do they even bother??"

they do it for a number of reasons.

1) TV time out front. the teams need exposure for their sponsors so by being out in front for a long break, they get that.
2) sprint or mountain points: the races within the race are important. by breaking away, those in the break can gobble up those points awarded on the days stages and try to set themselves up for a steal of that particular jersey

there are other reasons based on team strategy as well and it just depends on the stage, who the leader is, who the attackers are, etc. there's more to the tour than what really gets shown on TV. there's a lot more strategy than we think about and the races within the race alter the dynamic as well

and when it comes down to it, breaks do succeed. not always but they do and those in the break will still try to win but the pelloton sure moves fast when they want to.