It would be rare, in my opinion, for average cost to be "better" than specific ID. The only advantage I can think of offhand is that average cost is ever so slightly easier than spec ID when you actually sell shares.
The only situation where I would see it being useful is if a taxpayer had already sold some shares at average cost basis and didn't want to (or wasn't able to) then switch to spec ID.
It used to be that when you used average cost and then sold, you were locked into average cost forever. Now, it seems you can switch from average cost to spec ID and, if you had already sold, the shares before the switch would retain their average cost basis and shares after the switch can use spec ID.
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The reasons I can think of to use a DAF instead of just direct gifting to charity:
1. For tax reasons, you want the full charitable deduction now and want to make the donations later.
2. You want the anonymity that comes with giving from a DAF. The charity doesn't get your name.
3. You've got a lot of charities you support, and using a DAF makes the mechanics easier than transferring a bunch of shares directly yourselves.