There's a standard statement on the wiki page for every raid (at least for awhile) that says:

"While overkill damage does not contribute to the raid boss's global HP, it will contribute to the damage bonus. This bonus caps at 4200, which translates to 420 extra raid coins."

It's easy to show that doing at least 999,999,999 damage does have a small benefit to the player. It would be hard to prove whether all that damage contributes to the global HP, though, or at least it was when raids were first introduced. We've had some power creep since then and I think we can be pretty sure of at least a few things.

My base assumptions and observations:

• We don't do more damage against the global HP than we do against the boss. It wouldn't make sense to add digits to an already ridiculously large number for it to be more than a 1:1 ratio.
• Every raid boss has 4m HP. I believe this has been true for every raid so far, I originally thought the Bahamut raid might be less, but it doesn't appear to be so.
• The top 1% don't clear the raid so much that they would significantly change how much everyone else clears it. This is based on the fact that I'm a regular player who uses basically all my orbs but doesn't refill and hasn't leveled up, and I'm around 1000th place. Looking at the top 100, only about the top 60 have cleared it 1.5x as much as me, the top 20 about 2x as much as me, the top 10 about 3x as much as me and the top 4 about 4x as much as me, and the top ranked player is clearing it about 8.5x as much as me. Altogether, it seems like the top 100 players do about as many clears as about 200 normally active players.
• For the last year or so, we've floated between level 36-37 for most of basically every raid. The minimum rate we'd need to make it to level 37 is to do 30 trillion global damage in less than 6 hours (about 5 trillion per hour), but we've typically done about double that rate for the first week, and a bit less than double the rate for the second week. For the first week of this raid, we were almost double that, typically doing almost 20 trillion damage per hour, whether we're at level 36 or 37.
• There are a little more than 100,000 players who have cleared the raid at least once. I created an account and played enough to unlock the vortex and clear the lowest raid level once, and that account is tied for around 105,000th place. An account that only cleared it once, and did it earlier in the first couple hours before the raid level was up, but we definitely don't have any players we could call active below that account.

If we can do 20 trillion global damage against the raid per hour on a sustained basis, like we did last week, we need to assume one of a few things:

• At 4 million damage per clear, we'd need to be doing 5 million clears per hour, which would mean having at least 5 million active players.
• Most of our damage comes from bot accounts.
• The overkill damage counts up to 999,999,999, and we have about 20,000 players achieving that on an hourly basis (or probably a bit less, but with other players still doing a lot of damage and a long tail of players doing enough damage.
• The overkill damage counts up to the damage cap, and most of our damage comes from about 9,000 players doing capped damage.

#1 is clearly not true, we just don't have even close to 5 million players (or even close) who have cleared the raid at all, much less every hour.

#2 doesn't seem like it could be true without affecting the leaderboard by a lot - they don't seem to be in the top 100, because as I mentioned, the top 100 doesn't seem to do more than about 200 normal people's raid clears. They could be below the crowd of one-time clears, but that's not how things looked back when bots were a regular thing for raids.

It's hard to really distinguish between #3 and #4 from observable evidence.

tl;dr - Some amount of overkill damage obviously affects the global HP. In fact, the vast majority of damage we do to the global raid HP is from overkill damage. It's probably time we stopped pretending we know it doesn't.